The Jesus-movement that originally attracted the name “Christian” was most likely seen - by its members, by mainstream Jews and by non-Jews - as a movement within Judaism. Thus a Christian was probably thought of as a type of Jew.
Pretty early on, though, the Jesus movement started to admit non-Jews as members. This wasn’t uncontroversial, and the controversy within the movement is to some extent documented in the NT. The main issue seems to have been whether participation in the Jesus movement involved conversion to Judaism, or whether one could be a Christian without adopting Judaism and, in particular, without observing the rigours of the Law.
For a short time, Christianity was a partly Jewish movement. There were Jewish Christians, mostly centred around Jerusalem, who were Jews by birth and who continued to observe the Law. They went to synagogue on the Sabbath (and may have had their own synagogues) and celebrated the eucharist on the following day. Their views were controversial within Judaism but I think mainstream Jews would have accepted that, yes, these Christians were indeed Jewish. I don’t think they were ever more than a minority within Judaism.
At the same time, you had non-Jewish Christians, who were not considered by themselves or anyone else to be Jewish, and who did not observe the law. And I think from pretty early on they dominated the Christian churches everywhere except Jerusalem.
OK. We get to AD 70 and the Jewish revolt and its brutal suppression by the Romans, culminating in the destruction of Jerusalem. At this stage the Jewish Christian church in Jerusalem pretty well disappears from the record. It’s members presumably die in the fighting, or are enslaved or exiled afterwards. (They’re Jews, remember.) Maybe they are absorbed into non-Jewish Christian churches elsewhere. Maybe they fall away from Christianity. We don’t know.
From this time on, the Jesus movement outside Jerusalem has a strong incentive not to identify as Jewish, since Jews are seen as rebellious, troublesome, vanquished, etc. Plus, Judaism slowly regroups and struggles to reinvent itself and find an identity without either a Temple or a functioning, priesthood, and part of this process is a hardening of the view that Christians are not Jews. So Judaism and Christianity come to be seen as two quite distinct groups, and this suits both of them.
What proportion of Jews ultimately embraced Christianity? We don’t know. It’s thought, though, that about 10% of the population of the empire was Jewish before the Jewish revolt, and there were thriving Jewish communities in cities throughout the Mediterranean region. In the decades following the revolt the size and prominence of this population shrinks dramatically, while the size and prominence of the Christian movement grows It’s possible - but this is no more than speculation - that many of the Jews did embrace Christianity. But, if so, this was an attempt to distance themselves from Judaism, so they would have had no desire to emphasise any Jewish continuity or inheritance in Christianity.