Short answer: they got mixed in.
Jews who remained Jewish by faith had rules against intermarriage which allowed them to main a distinct ethnic identity even after the end of any kind of Jewish state. Jews who became Christian by faith on the other hand, had no reason not to intermarry with the Gentile Christians who fairly rapidly became the majority surrounding population. In certain areas (like the Church of the East and Ethiopia), the Jewish cultural component remained stronger in the resultant Christian communities than in others, but in all areas the Jewish and Gentile Christians simply became one people.
Any Jewish Christians as a distinct community in the present day would presumably be fairly recent converts (a couple of generations at the most) since anywhere outside of Israel, the same factors that caused the mixing in the first centuries of the Church would continue to apply.