So the premise here is that the Late Night Wars don't happen and Letterman stays on at NBC, until he finally takes over for Jay Leno. What happens to Conan O'Brien?
Probably moves and continues up through the writing ranks, maybe shifting to film, without ever getting any air time, maybe moving to doing voice work. Maybe a writer for Robot Chicken, but probably involved with Adult Swim.
The only way Letterman stays at NBC is if he gets the Tonight Show in 1992.
NBC gave him a calculated insult because they thought they could retain him in the Late Night slot. CBS or ABC will give him a slot against the Tonight Show, full stop.