First, I'll get the theological implications for Christians out of the way first. There is actually no theological requirement for Jesus to die violently to fulfill His mission, as long as He is born and dies just like any other mortal man. Nor for that matter for Jesus to be a man, though He has to be one sex or the other, or to be descended from David, become famous, found a world religion, etc.
Historically, provided Jesus goes to Jerusalem eventually during Passover and challenges the High Priests directly, the High Priests will ask the Romans to execute him for blasphemy. ACTS records that the Romans saved Paul in a similar situation by getting him exiled to Rome, but Roman was a Roman citizen and my guess that the provision that the Romans do the executions was put in place as a means to protect Roman citizens. The Procurators otherwise would execute any local troublemakers the local elites wanted them to. Pilate refusing to do this, on the grounds he was working towards in John, that Jesus hadn't actually done anything illegal (according to Roman law), would have advanced the Jewish rebelling thirty years later and ruined Pilate's life. A better POD is that Jesus comes to Pilate's attention earlier, and Pilate incorporates him into his household in Caesarea, in which case Jesus would not go to Jerusalem.
Either way, Jesus winds up in Rome, since he is making himself unpopular with the elites in Palestine, and if the Roman officials on the spot decide to save him they know they have to get him away from Palestine. They did this with Paul, but they had strong legal grounds since Paul was a Roman citizen who was exercising his right to appeal. Jesus is brought to Rome for further questing something something. Also, Roman officials would have seen through claims that it was Jesus who was planning a revolt against Roman rule in seconds. They would execute Jesus to keep things quiet or get him to Rome if they had an attack of conscience.
(Pilate, by the way, was Procurator, an official charged with looking after the Emperor's private interests, not a Propraetor or Proconsul, official representatives of the Magistrates, and certainly not the "Governor"of Judea as it is usually mistranslated)
After Jesus gets to Rome, he either remains obscure, though he might be mentioned if the action does spark a Jewish revolt, but he also might become famous as a philosopher like Epicetus. You could even get a religion. I could see Nero killing him, but the OP posits that he dies of old age. Also, OTL Jesus spent his entire adult life not just in Palestine but within walking distance of the Sea of Galilee so the experience of being in Rome would change him and his teachings, which seem to have been affected by the few meetings he did have with non-Jews.