One key factor determining the probability that Jesus actually died on the cross is the probability (or improbability) of the following claim:

(NTC) Jesus’ hands (or arms) and feet were nailed to the cross.

Crucifixion does not necessarily involve nailing the victim to a cross, and the Gospel accounts of the crucifixion don’t indicate how Jesus was attached to the cross. Binding the victim to the cross was more common than nailing. Furthermore, nails were probably used for crucifixions when there would be no guards keeping watch over the victims, but the three synoptic Gospels indicate that Roman soldiers crucified Jesus and then remained at the site of the crucifixion, thus making the use of nails even less likely in this case. The Fourth gospel agrees that Roman soldiers crucified Jesus, but is unclear about whether they remained at the site of the crucifixion.

It usually took days, not hours, for a crucified person to die, but Jesus was only on the cross for a few hours. Since the use of nails in crucifixion would cause the victim to die more quickly, the claim that Jesus’ hands (or arms) and feet were nailed to the cross (NTC) is an important part of the case for the belief that Jesus died on the cross.
1. Crucifixion does not necessarily involve nailing the victim to a cross.The Oxford Companion to the Bible has an article on ‘Crucifixion’ by Otto Betz. The article begins with a definition:
Crucifixion. The act of nailing or binding a person to a cross or tree, whether for executing or for exposing the corpse.(Oxford Companion to the Bible, p. 141)

Evangelical NT and Jesus scholar Craig Evans describes crucifixion this way:
Jesus was put to death by crucifixion, a form of execution practiced in late antiquity, whereby a person was tied or nailed to a pole or cross. To be crucified is, literally, to be “staked.”(Jesus, The Final Days, by Craig Evans and N.T. Wright, edited by Troy Miller, p. 28)

The Eerdmans Dictionary of the Bible has an article on ‘Crucifixion’ by John Donahue:
A particularly horrible mode of punishment by which a person (or sometimes the corpse of an executed victim) was nailed or bound to a cross…or to a stake or tree.
Crucifixion (from Lat. crux, “cross,” and a form of the verb figere, “attach” or “fasten”) was widely practiced in antiquity.(Eerdmans Dictionary of the Bible, edited by David Freedman, p.298)

The Dictionary of Jesus and the Gospels has an article “Death of Jesus” by Evangelical NT scholar Joel Green. Here is a comment from the article about the use of nails in crucifixion:
In the Roman world, however, the form of crucifixion was apparently more uniform: it included a flogging beforehand, and victims often carried the crossbeam to the place of crucifixion, where they were nailed or bound to the cross with arms extended, raised up, and perhaps seated on a sedicula, or small wooden peg (Hengel 1977, 22-32).(Dictionary of Jesus and the Gospels, edited by Joel Green, Scot McKnight, I. Howard Marshall, p. 147)
2. The Gospel accounts of the crucifixion don’t indicate how Jesus was attached to the cross.
Evangelical NT scholar Gerald Borchert also comments on the use of nails in crucifixion:

The way the hands and legs were fastened was normally either by tying the limbs with ropes or by fastening the limbs with nails. None of the crucifixion stories in the Gospels actually indicate how Jesus was attached to the cross…
(The New American Commentary, Volume25B, John 12-21, p.263)
3. Binding the victim to the cross with ropes was more common than nailing.Evangelical NT scholar Craig Keener discusses how victims of crucifixion were attached to the cross:
Executioners usually tied victims to the cross with ropes but in some cases hastened their death by also nailing their wrists…(The Gospel of John, Volume 2, p.1136)

Evangelical NT scholar Darrell Bock comments on how Jesus was attached to the cross:
The cross [of Jesus] stood about seven feet high and, since the crossbar was either at the top or just below on the vertical pole, looked like a T or a t . He [Jesus] would have been held on the beam by rope or by nails, though the latter was less frequent.(Jesus according to Scripture, p. 385)

If tying a victim to the cross was more common than nailing, then the prior probability that Jesus was nailed to the cross is less than .5. A reasonable estimate of the prior probability that nails were used in Jesus’ crucifixion is
.3 or .4.

But NTC claims more than just that nails were used in Jesus’ crucifixion. It claims that nails were used to attach Jesus’ hands (or arms) to the cross, AND that nails were also used to attach Jesus’ feet to the cross. In some crucifixions, nails were used for attaching the hands or arms but not the feet, and in some crucifixions nails were used for attaching the feet but not the hands or arms. So, the prior probability of NTC is even lower than .3 or .4. A reasonable estimate of the prior probability of NTC being true would be: .2 (based on the assumption that about half the time when nails were used, they would be used on both the hands (or arms) and the feet of the victim).

Note that in the one and only case of physical evidence of a crucified Palestinian man, who was crucified in the first century, nails were used to attach his feet to the cross, but apparently not used to attach his hands or arms to the cross. (See “Death of Jesus” by Joel Green in the Dictionary of Jesus and the Gospels, p.147-148).
4. Nails were probably used for crucifixions when there would be no guards keeping watch over the victims.Evangelical NT and Jesus scholar Ben Witherington comments on the motivation for the use of nails in crucifixion:
The reason for the nails seems to be the prevention of escape, for in noncelebrity crucifixions or during a war there would frequently not be a guard, and often persons lived for a good while, sometimes long enough to be taken down from a cross, especially under cover of darkness. The cross would sometimes be only a few feet off the ground, allowing friends to approach the victim and attempt a rescue.(The Gospel of Mark, p.395)
5. The three synoptic Gospels indicate that Roman soldiers crucified Jesus and then remained at the site of the crucifixion.
MARKIt was nine o’cloc

k in the morning when they [‘the soldiers’ see v.16] crucified him [‘Jesus’ see v.22].(Mark 15:25, NRSV)At three o’clock Jesus cried out with a loud voice…(Mark 15:34, NRSV)Then Jesus gave a loud cry and breathed his last. …Now when the centurion, who stood facing him, saw that in this way he breathed his last…(Mark 15:37-39, NRSV)

Note:
A centurion …was a professional officer of the Roman army after the Marian reforms of 107 BC. Most centurions commanded 83 men despite the commonly assumed 100…
https://en.wikipedia.org/wiki/Centurion
MATTHEWAnd when they [‘the soldiers of the governor’ see v. 27] had crucified him, they divided his clothes among themselves by casting lots; then they sat down there and kept watch over him.(Matthew 27:35-36, NRSV)
From noon on, darkness came over the whole land until three in the afternoon. And about three o’clock Jesus cried out with a loud voice…(Matthew 27:45-46, NRSV)
Then Jesus cried again with a loud voice and breathed his last. At that moment the curtain of the temple was torn n two, from top to bottom. The earth shook, and the rocks were split. … Now when the centurion and those with him, who were keeping watch over Jesus, saw the earthquake and what took place, they were terrified…(Matthew 27:50-54, NRSV)
LUKESo Pilate gave his verdict that their [‘the chief priests, the leaders, and the people’ see v. 13] demand should be granted [for Jesus to be crucified]. …and he handed Jesus over as they [‘the chief priests, the leaders, and the people’] wished. As they [unclear reference] led him [Jesus] away, they [unclear reference] seized a man, Simon of Cyrene…and they [unclear reference] laid the cross on him [Simon], and made him carry it behind Jesus.(Luke 23:24-26, NRSV)

Commentary on the unclear reference of ‘they’ in verse 26 by NT scholar John Nolland:
Does Luke want us to understand that Jesus is in Jewish or Roman charge at this point? In strict grammar, the ‘they’ who lead Jesus out to execution ought to be those who have called for his execution, and Luke has certainly wanted to emphasize the Jewish desire for this outcome (v 25). But the natural reading of vv18-25 leaves the crucifixion in Pilate’s realm and therefore in Roman hands. Also the continuing Lukan text provides no encouragement for viewing the execution proceedings as in any sense in the hands of the Jewish people or leaders. Since already in vv 14, 18 Luke has shown a certain carelessness about grammatical antecedents, it seems best to retain the consistency of the Lukan picture by assuming the same level of grammatical carelessness here…(Word Biblical Commentary, Volume 35c, p.1136)

Side comment: It is a bit hard to believe that the very words of Luke’s gospel were inspired by an all-knowing being, when the carelessness of Luke’s grammar helped to contribute to the view of the Jews as ‘Christ killers’ which in turn led to the murder of six million innocent men, women, and children in the 20th century. Wouldn’t an omniscient deity have been aware of the potential for future prejudice of Christians against Jews, and thus helped Luke to write grammatically correct sentences in order to avoid promoting such prejudice?

As hinted by Nolland’s commentary, Roman soldiers continue to be a part of the crucifixion scenes in Luke, thus confirming Matthew’s claim that the soldiers remained near the cross to watch over Jesus and the other men crucified with him:
Two others also, who were criminals, were led away to be put to death with him [‘Jesus’ see v. 28]. When they came to the place that is called The Skull, they crucified Jesus there with the criminals…. And they cast lots to divide his clothing.(Luke 23:32-34, NRSV)

While it might be conceivable that Pilate would hand one despised public figure over to a Jewish crowd to be killed, it is very unlikely that he would hand over the execution of various other condemned criminals who have violated Roman law, to be executed by an unruly Jewish crowd. Plus the casting of lots to divide Jesus clothing is something that the gospels of Mark and Matthew have Roman soldiers do, and that action makes more sense if we are talking about Roman soldiers. Thus, this passage implies that the crucifixion was carried out by Roman soldiers in Luke’s view.

Luke has the Roman soldiers remaining on location after Jesus is attached to the cross:
The soldiers also mocked him, coming up and offering him sour wine…(Luke 23:36, NRSV)
It was now about noon, and darkness came over the whole land until three in the afternoon, while the sun’s light failed; and the curtain of the temple was torn in two. Then Jesus, crying with a loud voice, said, “Father, into your hands I commend my spirit.” Having said this, he breathed his last. When the centurion saw what had taken place, he praised God and said, “Certainly this man was innocent.”(Luke 23:44-47, NRSV)

An officer of the Roman soldiers is present at the alleged death of Jesus in the afternoon, again confirming the view that Roman soldiers remained at the site of the crucifixion to ensure that nobody attempted to rescue Jesus or any of the other crucified persons.

As with the other Gospels, Luke has Joseph of Arimathea request Pilate for the body of Jesus:
This man [Joseph of Arimathea – see verses 50 & 51] went to Pilate and asked for the body of Jesus.(Luke 23:52, NRSV)

If Luke believed that Pilate had handed Jesus over to ‘the chief priests, the leaders, and the people’ (mentioned in Luke 23:13) for them to carry out the crucifixion, then it would make no sense for Luke to have Joseph go to Pilate for permission to remove the body of Jesus from the cross. If Jesus was crucified by Jews, then Joseph would have gone to the chief priests for permission to remove the body. So, this verse about Joseph of Arimathea implies that Jesus was crucified by Roman soldiers and that Jesus was still in the custody of those soldiers on the afternoon of the crucifixion.
6. The Fourth gospel agrees that Roman soldiers crucified Jesus, but is unclear about whether they remained at the site of the crucifixion.
JOHN
When it comes to the crucifixion of Jesus, John use of the pronouns “them” and “they” is even more problematic than Luke’s:
…He [‘Pilate’ see v. 13] said to the Jews, “Here is your King!” They [‘the Jews’] cried out, “Away with him! Away with him! Crucify him!” Pilate asked them, “Shall I crucify your King?” The chief priests answered, “We have no king but the emp

eror.” Then he [‘Pilate’] handed him [‘Jesus’ see v. 13] over to them [‘the chief priests’ and/or ‘the Jews’] to be crucified.(John 19:14-16, NRSV)

Unlike the Gospel of Luke, I suspect that verse 16 intentionally refers to ‘the chief priests’ and ‘the Jews’ in order to portray ‘the Jews’ as evil Christ-killers. The Gospel of John is sometimes called the ‘Gospel of Love’ but it is really the gospel of hatred and anti-semitism, or more accurately, the gospel of anti-Judaism.

Near the close of the first century, the Jewish followers of Jesus were being excommunicated from Jewish synagogues, and the Gospel of John reflects the anger and hatred of Christian believers towards ‘the Jews’ who were kicking them out of their religious communities. This anger and hatred is then projected back into the words and teachings of Jesus, so that, for example, there is no mention of the following loving teachings of Jesus in the gospel of John:
Love your enemies.Pray for those who persecute you.Turn the other cheek.Forgive others, so that God will forgive you.Do not be angry with your brother.Be merciful, as your heavenly Father is merciful.Do for others what you would have them do for you.

Judge not, lest you be judged.

Remove the beam from your own eye, and then you will see clearly to remove the speck from your neighbor’s eye.
Instead of Jesus preaching love, tolerance, mercy, and forgiveness, we get the nonsensical picture of Jesus, a devout Jew, going into an angry tirade against ‘the Jews’:
They answered him, “Abraham is our father.” Jesus said to them, “If you were Abraham’s children, you would be doing what Abraham did, but now you are trying to kill me, a man who has told you the truth that I heard from God. This is not what Abraham did. You are indeed doing what your father does.” They said to him, “We are not illegitimate children; we have one father, God himself.” Jesus said to them, “If God were your Father, you would love me, for I came from God and now I am here. I did not come on my own, but he sent me. Why do you not understand what I say? It is because you cannot accept my word. You are from your father the devil, and you choose to do your father’s desires. He was a murderer from the beginning and does not stand in the truth, because there is no truth in him. When he lies, he speaks according to his own nature, for he is a liar and the father of lies….Whoever is from God hears the words of God. The reason you do not hear them is that you are not from God.” The Jews answered him…(John 8:39-48, NRSV)

These are not the words of the preacher of love, tolerance, mercy and forgiveness. These are the angry and hateful expressions of Christian believers who were being excommunicated from Jewish congregations several decades after Jesus was crucified. Here is a footnote on 8:44 from my HarperCollins Study Bible:
This harsh language indicates the intensity of the conflict between the Jewish Christian community for which John was written and the synagogue authorities…(The HarperCollins Study Bible, NRSV, p.2030)

Side comment: Thus, the “gospel of love” was a significant contributing factor in the murder of millions of innocent human beings in Nazi Germany, because the angry anti-Judaism of John evolved into the hateful anti-semitism of Germans in the twentieth century. Again, I don’t see how an omniscient person could be unaware of the potential for the angry anti-Judaism of the words attributed to Jesus in the Fourth gospel to generate prejudice and hatred between Christian believers and Jews. This is a good reason to doubt the divine inspiration of the Fourth gospel.

Nevertheless, the Fourth gospel also indicates that Jesus was actually crucified by Roman soldiers, not by ‘the Jews’:
When the soldiers had crucified Jesus, they took his clothes and divided them into four parts, one for each soldier. …(John 19:23, NRSV)

One comment in the Fourth gospel suggests that at least some of the soldiers left the crucifixion site and returned later to finish off the victims:
…they [‘the Jews’ see v. 31] asked Pilate to have the legs of the crucified men broken and the bodies removed. Then the soldiers came and broke the legs of the first and of the other who had been crucified with him [‘Jesus’ see v.30].(John 19:31-32, NRSV)

The Fourth gospel does not explicitly state that all of the Roman soldiers left the site of the crucifixion, and even if that is what the author intends to imply by the above passage, the opposing accounts of the synoptic Gospels would outweigh the less reliable claims of the Fourth gospel. So, it is very likely that at least some of the Roman soldiers remained at the site of the crucifixion, and could thus prevent any attempt at a rescue of the crucified men. Thus, there would not be the standard motivation for the use of nails in this case (i.e. to prevent rescue of a crucified person while the victim was left unguarded).
7. It usually took days, not hours, for a crucified person to die.The Dictionary of Jesus and the Gospels has an article “Death of Jesus” by Evangelical NT scholar Joel Green. Here is a comment about crucifixion from the article:
Among the torturous penalties noted in the literature of antiquity, crucifixion was particularly heinous. The act itself damaged no vital organs, nor did it result in excessive bleeding. Hence, death came slowly, sometimes after several days…(Dictionary of Jesus and the Gospels, edited by Joel Green, Scot McKnight, I. Howard Marshall, p. 147)

Eerdmans Dictionary of the Bible has an article on ‘Crucifixion’ by John Donahue. This is a comment from the article about how long it takes for crucified people to die:
Often crucified people lingered for days, and death came ultimately from loss of blood or asphyxiation.(Eerdmans Dictionary of the Bible, edited by David Freedman, p.299)

Evangelical NT scholar Craig Blomberg comments about how long it took for a crucified person to die:
Death [by crucifixion] usually proceeded very slowly, often over a period of several days… Jesus died unusually quickly…(Jesus and the Gospels, p.346)

Evangelical NT scholar James Brooks discusses how long it took for a crucified person to die:
Death usually came slowly as a

result of exposure and exhaustion. Inasmuch as no vital organ was damaged, it often took two or three days for the subject to die…(The New American Commentary, Volume 23, Mark, p.256)

James Brooks comments on Mark 15:44 (“Pilate was surprised to hear that he [Jesus] was already dead…”):
Pilate was surprised because it often took two or three days for crucified persons to die.(Mark, p.266)

Evangelical NT scholar William Lane comments on Mark 15:44:
He [Pilate] was surprised, however, that Jesus was already dead. Since contemporary records show that crucified men often lived two or three days before dying, there was something extraordinary about the rapid death of Jesus.(The Gospel According to Mark, p.579)

Evangelical NT and Jesus scholar Craig Evans comments on Mark 15:44:
Bultmann is probably correct that the draft of Mark used by Matthew and Luke did not contain this material [vv 44-45], which may have been added later to explain how it was that permission would have been given to take down a crucifixion victim the very day that he had been crucified—indeed, after hanging on the cross only a few hours. After all, those crucified normally suffered for days.(Word Biblical Commentary, Vol. 34B, Mark8:27-16:20, p.516)

Evangelical NT and Jesus scholar Craig Evans on how a crucified person can take days to die:
Normally crucifixion victims were left to die, however long that took (sometimes several days).(Jesus, The Final Days, by Craig Evans and N.T. Wright, edited by Troy Miller, p. 31)

Evangelical NT and Jesus scholar Joel Green comments on how long it took for crucified people to die:

Bound or nailed to a stake, tree, or cross, the victim faced death with all organs intact and with relatively little blood loss. As a consequence, death came slowly, sometimes over several days…
(The Gospel of Luke, p.810)
8. The use of nails in crucifixion would cause the victim to die more quickly.Evangelical NT scholar Craig Keener discusses how victims of crucifixion were attached to the cross:
Executioners usually tied victims to the cross with ropes but in some cases hastened their death by also nailing their wrists…(The Gospel of John, Volume 2, p.1136)

Evangelical NT and Jesus scholar Ben Witherington comments on the use of nails in crucifixion:
Nails were often used for crucifixion, and of course this means of impaling a person on a board also caused blood to flow and so hastened death. In all probability what Jesus was expected to carry was the crossbar of the cross, which, once one was impaled on it, was dropped into a slot in the vertical beam which was already set in the ground. At that point the person’s feet would be secured either with nails or ropes.(The Gospel of Mark, p.395)

Evangelical NT scholar Craig Blomberg comments about how flogging and the use of nails could result in a quicker death for a victim of crucifixion:
Death [by crucifixion] usually proceeded very slowly, often over a period of several days… Jesus died unusually quickly, perhaps because of his previous flogging and perhaps because of the use of nails rather than ropes.(Jesus and the Gospels, p.346)

Note that since the question at issue for us is ‘Did Jesus die on the cross?’ one cannot assume that Jesus died on the cross and then infer that his hands and feet were nailed to the cross. That would beg the question at issue. The reasoning has to go in the other direction. First one must determine whether NTC is true (or the degree of probability of NTC) and then one can use that as part of the basis for drawing a conclusion about whether it is true that Jesus died on the cross (or to determine the degree of probability that Jesus died on the cross).