It is difficult to say what could have happened. This is just what I think would happen if the courier never got shot.
If the courier was never shot in the first place, then he would still be in the Mojave Express and Ulysses could still be luring him into the Divide. As for the conflict in the Mojave, the Omertas would probably be successful in their coup and Heck Gunderson's son would have been eaten. As for Benny...well he is still dead either anyways, but the bad side to that is no one will be able to get the Platinum Chip and activate the Securitron Army. Boone wouldn't have gotten to his senses after the Bitter Springs massacre, Raul would still be in trapped at Black Mountain, the State of Utobitha would still be flocking with Super Mutants and Nightkins. Also there could be a fullblast riot at Freeside and The Follower of the Apocalypse would have been overwhelmed by the amount of patients.
And the Brotherhood of Steel would have died in the Hidden Bunker.
Goodsprings would have given up Ringo too, Primm would still be in crap-state as Beagle is still captured, NCR is frozen shit, and nobody would be able to reprogram Primm Slim.
Novac would continue getting Ghouls from Repconn and Jason Bright's journey would have come to a complete halt. The NCR would eventually discover the power of ARCHIMEDES at Helios One and Jessup's gang would have been killed off by the Monroe's squad.
So many things would change if the courier hadn't been shot in the head.