Really interesting topic actually but most early 'Christians' didn't really think of themselves as converts but rather just Jews who understood Jesus to represent the 'completion' of Jewish script and prophecy.
Best example is Paul who most definitely continues to view himself as a Jew.
Anyone interested should check out a book like 'Did Jesus Exist?' by Bart Erhman or a Marginal Jew (huge read). There's a better book by him on the topic but blanking on the title