What happened to change Joe's timeline?

I've thought about that too. It's not explicitly explained how time travel works except for how Old Joe explained how his memories change and were a "sea of probabilities" until Young Joe formed new ones.

My thoughts are it's the infinite worlds hypothesis, there are an infinite number of realities, maybe old Joe only had a 5% chance of overwhelming his captors and removing the hood, giving Young Joe a split second's hesitation so he can avoid having his loop closed. We're only shown two different "loops" in the movie.

If you're looking for the "trigger" my idea is that the phone call that gives him the "Rainmaker"'s number is the trigger. Now he has something to look for in the past, and a slim hope of avoiding his fate…

There were no changes in Joe B's timeline. His wife WAS shot the first time (as Joe A) but he could do nothing, was captured (as shown when he was shown sprawled across the floor) and sent back and killed. The second time, he managed to over power his captors and send himself back. This was the change in Joe A & B... Also, Joe B knew Joe C would shoot him as soon as he arrived, hence he did the turn back, throw brick, punch and escape thing...

I may be completely wrong here, but . .

I personally thought that a paradox had been created in that timeline. When Young Joe fell from the fire escape I thought the fall killed him, which would have stopped him living to be old joe, to come back & die, thus negating the need for young joe to go on the run in the first place . . .

I took this as the break in the timeline that allowed us to move to another one where young joe lives on to become old joe, who gets sent back and escapes.

I do love a good time travel brain teaser.