Jesus probably did not have a name for the whole idea of a religion that has a name. He would have said that he and the apostles were Jews (yehudim in Hebrew), but the abstract noun form Judaism (yahadut) wasn't use until later. Similarly, he would have said that Pilate was a Roman, and that he worshipped idolatrous Roman gods, but there wasn't an "-ism" kind of name for that either.

After Jesus's death, his followers started to call themselves Christians relatively quickly. The Acts of the Apostles was written around 115 uses the word "Christian" for the followers of Jesus, so no later than that but maybe a little earlier. However, it took longer before Christianity became the name of a religion and not just a belief system. For a while, it was more like someone saying they're an Aristotelian or something like that-- it means they believe in Aristotle's teaching, not that they are part of a defined organized religious group based on Aristotle.