The problem with a lot of this discussion is that it's based on writings by the groups that won, but which do not show you what led up to it. A good source would be "The Ways That Never Parted: Jews and Christians in Late Antiquity and the Early Middle Ages", ed Adam H. Becker, Adam H. Becker, Annette Yoshiko Reed. The picture one gets from recent work is that up to Constantine, Judaism was a lot more varied than the Talmud and the catholic and proto-catholic writers imply. In many cities, Jews took an active role in the city, which would involve sacrifices to the city gods. This was not viewed as a problem by many, maybe even most Jews. Neither was Christianity. Similarly, many Christians saw no problem with observing the Sabbath, and joining in various Jewish activities.
That makes it hard to define exactly what "Jewish Christianity" was. Probably the real definition would be ethnic Jews who accepted Christ in one way or another. The implication one often gets is that this was a small remnant of James' people from the first century, who slowly died away. But with the large overlaps that actually occurred between people who hung around the synagogues and churches, that would be misleading.
While most remaining literature is from the people who wanted to purify Judaism and Christianity, there are remaining books that one would normally classify as Jewish, but which also accept Christ in one way or another, e.g. the Pseudo-Clementines.
What I've been reading recently suggests that what finally caused a split was when the Roman government adopted Christianity, and supported its leaders' attempt to purify Christianity.
