Seeing as Caesar’s Legion is still around in the Mojave in 2296, it seems safe to say the NCR’s campaign there didn’t fare so well in canon. Which makes sense, because the NCR was doing terribly and they would’ve gotten stomped by the Legion if the Courier wasn’t around to solve all of their problems for them (Picus the Legion spy blowing up the monorail and feeding the Legion info, false reports from Chief Hanlon, no help for Bitter Springs or Forlorn Hope, and much more). Seems likely that the Legion won the second battle of Hoover Dam, and Shady Sands being nuked certainly wouldn’t have helped matters either.

No Courier means House doesn’t get his Platinum Chip either, because Benny would get caught trying to sneak into the Fort and it would end up in the Legion’s hands who would rather destroy everything in the Securitron vault, so no Independent or House endings either. Whether Caesar is still alive or not, however, remains to be seen because the Courier is responsible for removing his brain tumour and without them, there’s nobody to grab replacement parts for the Auto-Doc or do the surgery in their stead.