It seems clear from the NT that the Jewish Christians didn't think Jesus had fulfilled the entire law of Moses - for example, they kept the laws relating to diet and circumcision and expected all Jewish converts to do likewise.
Before the Jerusalem Council (Acts 15) they (or at least some of them) believed Gentile converts must also keep those Mosaic laws.
In Acts 21, James - speaking on behalf of Jewish Christians who "are all zealous for the law" - is concerned by the rumour that Paul is teaching "all the Jews who are among the Gentiles to forsake Moses, telling them not to circumcise their children or observe the customs".
In Galatians 2, Peter didn't want to be seen eating with the Gentile Christians because he feared "the circumcision party" - Jewish Christians visiting from Jerusalem who were under James, who would have come down heavily on Peter if they saw him eating "unclean" Gentile food.
Paul accused Peter, Barnabas (a Jewish Christian) and the visiting Jews of acting "insincerely" because they evidently began insisting that the Antioch Gentile Christians "live like Jews” (Gal 2:14).
Paul, who vehemently opposed circumcision for Christians, had Timothy - a Christian - circumcised "because of the Jews". Timothy's mother was Jewish, so the Jews would have considered him Jewish by law. (
Acts 16:1-3)