I wouldn’t put it quite that way, but in broad strokes you’re right that the Jews converted to Christianity. I was a member of an Antiochian Orthodox Christian parish in a major US city, with people from the old country (middle eastern countries) arriving regularly, and that place was full of Jewish noses, because yes, that’s what God’s people did when they met the Messiah and the ministry of the Holy Apostles. It’s recorded in the book of Acts and the epistles.