Where I'm from, when an offshoot timeline happens, the prime one keeps going as normal. Think of it as there being two parallel timelines where everything is going exactly the same in each, but in one you suddenly take a sharp left instead of going straight. So suddenly, you're no longer the prime timeline and the other one now is. So the new prime timeline will keep going and everything that you were going to do in that one happens there.

[ he has no idea if that makes sense or not but it's really been a while since he has had to explain how different timelines might work... ]

I guess another possibility is that we just stop existing in that other timeline altogether? Like we were just plucked out of there and put into this one.