From what I can see of the four primary gospels, only the Gospel of John written around the turn of the 2nd century actually mentions using a nail during the crucifixion whereas the other earlier gospels do not.
What is the likelihood that the use of a nail in the Gospel of John could have been influenced by the Siege of Jerusalem in 70 A.D. or the work done on it by the historian Josephus whom mentions that the Romans mostly nailed Jews to crosses during it?
Essentially what is the likelihood of the historical Jesus being nailed to the cross against the seemingly more common tying and did Josephus influence later accounts?
Nails were not unusual in Roman crucifixions. In a 2018 report on a possible crucifixion victim recovered in northern Italy, the authors state that "The [crucifixion] method preferred by the Romans did not involve tying the condemned person to the beam...but rather the use of nails." The authors are reporting on a skeleton recovered in the Po valley, male, early 30s, with a hole through his heel bone probably caused by a crucifixion nail. (See the article for photos, and also their original sources in the Bibliography.)
Crucifixion nails are rarely found in excavated skeletons, but there's a famous example from Israel of a first-century Jewish man whose ankle bones were recovered with the nail in place. This article from the Times of Israel has good photos, although it's not correct when it states that this is the "sole physical evidence" ever found for the practice of Roman crucifixion. It’s a rare (perhaps unique) example of a nail still embedded in the bone, but there have been other skeletons found with trauma to ankles and arms suggesting use of crucifixion nails.
As for why the nail isn't mentioned in three of the four gospels, I'll leave that to Bible scholars. Given that the use of nails was a common Roman practice, it seems likely that it was ommitted because it was ordinary enough not to comment upon.