The reason it was switched among Jews from just snipping the tip to radical circumcision is because Hadrian banned circumcision after the Bar Kokhba rebellion (or maybe after the Kitos War, I forget exactly) and the ban was not lifted until the reign of Marcus Aurelius (if I recall my history correctly).
So, many Jewish men were able to pass themselves off as Hellenic (say in the baths or the gymnasia), because if you didn't get a good look you couldn't tell exactly. The shift to radical circumcision was to ensure that Jewish men could not do that; that there was no way to make it appear as if they were uncut.
As the Byzantine Christians took power and persecuted the Jews even more intensely than had the Romans, the custom became all but universal for the same reasons. So one could not hide it. And that has more or less remained the custom to this day, although there have been periodic movements to go back to the less invasive procedure.
The point of it of course was to seal the covenant of Abraham, and it was not thought that you had to rip the whole prepuce off in order to do that. Even with the shift to radical circumcision it was not a matter of the covenant so much as a cultural move to distinguish the Jewish community from their persecutors.
I don't know the precise history with regard to Islam, though it's worth noting that it's not in the Koran but rather in one of the Hadith, which are alleged sayings of Mohammed (probably made up) not in the Koran.
Interesting. So, are you saying the Romans and Byzantines forced complete circumcision on the Jews?