a new pope being elected does not cause the crusade timer to get reset.
I'm not 100% on this but it seems like it takes 20 turns and nothing else.

My story to back up my claim.
-I had a successful crusade to Jerusalem in the 20 something turn.
-10 turns later one of my cardinals became the pope.
-From the first crusade i was checking if i could launch a new one every 2-3 turns.
- in the 40 something turn i was able to request a new crusade.