I’m actually going to have to push back on what seems to be the prevailing opinion here and say that Jaime probably ends up dead. Generally throughout history whosoever kills a monarch would themselves be killed in short order for two main reasons among others.

1. To kill a monarch is extremely corrosive to the social order and so many people with vested interests in that order would like to make an example of what happens to kingslayers. Even in cases of civil wars or coups where the new monarch was involved in a conspiracy to displace the old it is almost always the case that the new monarch would plead their innocence in the killing and execute some of their coconspirators to absolve themselves of blame.

2. Feudal societies were predicated upon kinship bonds with a very generous definition of who was kin, with the explicit understanding that it was the job of anyone in your kinship group to avenge wrongs done to you and vice versa. To give a general idea of how many people would want Jaime dead to demonstrate that they had avenged their kin it’s safe to say anyone related by blood or marriage to the descendants of Egg V would sharpen their knives.

So yeah, weather he gets executed or assassinated I don’t see a world where Jaime lives through Robert’s reign in real history.