Well then someone should have told all those Jews that came from the area of that caravan stop and travelled all the way to Jerusalem for the day of Pentecost in the Book of Acts. All the sarcastic comments in the world can't change the historical fact that Babylon was in ruins by the time the NT was written, and the fact that the Iraqi church has no tradition of being founded by St. Peter.
None of the "Babylonian Talmud" was written in the city of Babylon, for the simple reason that it was not a city anymore at that time. The rabbinic schools from which it came were called, generically, "Babylonian," because they were in Iraq.
A good overview of the history is here. It lists the individual schools and the towns where they were located, which don't include Babylon.