A letter to the Church in Galatia suggest that Paul had a serious disagreement with Peter and the argument started as Peter felt that Christianity was a new form of Judaism, and if you converted you had to become a Jew. Paul disagreed, of course.
The record of how it was worked out was not recorded. Peter moved toward full acceptance of gentiles into The Way, but how far he moved, no one knows. I would express only Paul's point of view, and I must conclude that his point of view prevailed at councils during the very early times because Paul was a minority voice that became the predominant one maybe through ordinary selection and the form of the faith that retained it Jewishness never really caught on, so its numbers dwindled, even though the Pauls type of the faith began to grow.
I know that there was a Jewish Christianity and for a while it survived, but it died out very early on.
The major difference between Judaism and Christianity still today is that Christians energetically support new members and go all-out to make it easy for new members to join. Jews have never tried to catch the attention of new members and do not make it particularly easy for new members to join. Simply put, Judaism is turned inward; Christianity is turned outward.