The NCR is all over the entire West coast. In New Vegas they mention probes into Colorado, so I would assume they were pushing east in the former North Western states as well. Not sure what's official though. So it would be hard for them to convincingly have Vault-Tec have that many nukes left to take out the whole Western side of the country. I think what we will find out is that they took out the capital (Shady Sands, President, cabinet whatever), and not all. The Shady Sands sign in the show even mentions "First Capital of the NCR", which implies the capital moved before it was nuked. The Vault with the cyclops overseer has a NCR flag, which implies they are still part of the NCR.