What if Rome had become Jewish?

A historical "what if" question.

Constantine adopted Christianity as the religion of the Roman Empire in the 4th century. But in prior centuries, Judaism was a massively proselytizing religion, in Rome and beyond. And though the historicity of this is doubtful, according to the Talmud the emperor "Antoninus" (identification uncertain) actually did embrace Judaism.

So here's the question. What if, instead of converting to Christianity, the Roman Empire had become Jewish?

What would the world look like today?

Would Judaism and Islam have been the great historical rivals, instead of Christianity and Islam? Would Islam even have existed?

What would Judaism look like?

Would a Jewish Rome have rebuilt the temple in Jerusalem? Or would they have shifted the centrality of the religion to Rome itself?

Would Judaism still be a religion of shuls and Torah learning? Or would it look very much like the Catholic Church, with grandiose Cathedrals and a pope-like figure in Rome, but with some version of halacha and a different dogma?

Would Hebrew still be out Lashon Kodesh? Or would it be Latin?

It's impossible to know any of this, of course. But it's a fascinating thought experiment.