Before you can have emperor-worship, you have to have an emperor. Having none, the Roman Republic never had a cult. This had to wait until the Romans had successfully destroyed their republic.
As the master of so many different nations, with so many different gods, the Roman emperor needed more than just the brute force of the Roman army–so often resorted to–to keep them all in line. So what the Romans did was to evolve an imperial cult; and everybody, regardless of their own national religion, had to swear oaths in the emperor’s name and perform sacrifices to him.
This was really inclusive. For a time, Jews got a pass: they prayed to their God for the emperor. Everybody else had to pray to the emperor. For anyone to do otherwise would have been divisive. The Romans were big on inclusiveness, and anyone who tried to be divisive would be killed.
Christians refused to sacrifice to the emperor. Once Nero figured out that the Christians weren’t just an eccentric Jewish sect, he initiated persecutions. The Christians’ failure to sacrifice to him was divisive. Even exclusionary. So he killed as many as he could. Other emperors followed in his footsteps.
Notice Nero’s image on the coin. In Republican times, no living person could have his picture on a coin. That was reserved for gods. So the parade of Roman coins with emperors’ portraits on them tells us where they were coming from.
In the long run, Jesus Christ, who had no army, prevailed against the Roman Empire, which did. He conquered it.
Let them who have ears, hear. Let them who have eyes, see.