My (somewhat liberal) Christian take.
First off, claims that Christ may not even have existed strains Occam's razor in many ways. One has to accept a conspiracy theory that Josephus was tampered with. More importantly, it is hard to accept that a small sect of Judaism would ever have spread so fast and become the state church of the Roman Empire in 350 years without some pretty strong experiences galvanizing its early founders. Since these experiences are all claimed by the early gospel writers to focus on a man named Jesus, it seems needlessly complex to claim (with absolutely no information) that he did not exist at all. It seems to me that skeptics are much better off accepting the basic premise that a dynamic and controversial prophet named Jesus existed who threatened the status quo and was executed by the Romans - and then question the particular claims Chistianity makes about him and his divinity.
"Passion" is not anti-semetic, period. It does take the Gospel claims about the Jewish priestly leadership's culpability at their face value - as well as the the biblical sympathetic portrayal of Pilate. However, throughout the film, Gibson makes the point of showing Jews (even some of the temple priests) opposing his crucifixion - and many Romans come of as brutish SS thugs much worse than any of the Jews. It may or may not be what actually happened, but it is a fair retelling of the gospels. I even found the alleged "bad guy" Caiaphus presented in a manner that you could understand.
Regarding the original question, what happens next all depends on what you believe Jesus was and what he actually believed about himself. If he did indeed believe he was the Messiah and his death was needed to absolve the sins of mankind, he and his disciples would eventually make sure he was put to death somewhere or sometime else (and we'd never bear about the Pilate story in the gospels). Christianity might still develop along much the same way as in OTL (at least that what Christians like myself have to believe).
If he wasn't the Messiah - or if the gospel writers put those words into his mouth after his death, then it's reasonable to assume Jesus would have accepted this banishment willingly. He and his version of Judaism might have taken hold in Parthia or some other place at the outer edges of the Roman Empire. If the sect survived and found converts, it is also reasonable to assume it might have gradually spread back into Judaea and the Empire after Jesus's death. It might or might not have reentered the mainstream of Judiaism, or developed - like Islam - as another Abrahamic monotheistic faith (The God of Abraham does have this annoying habit of trying to make people believe in him! - oops sorry). It might have turned Judaism into an expansionist evangelistic religion, replaced it, or competed with it and other cults for Roman citizens. I doubt very strongly if the Jesus cult would have become dominant, however, because without the particular claims Christianity makes about Christ, it would have been not all that differenet from any other Jewish sect (and probably would not have reached out much to gentiles).
Just thought I would help bust the stereotype.