I think the whole thing would be settled very shortly after the Legion takes the Mojave. Once they punch through the mountains, the next target is the Hub in the middle of the California High Desert, on a direct path from Long 15. The Hub, as it tied the Wasteland together in Fallout 1, ties together NCR, dividing it in three. To the north, you have the political center of the State of Shady (along with Junktown). To the south is the industrial capital in the Boneyard, along with Dayglow. To the west is the presumptive agricultural center, the more-recently acquired Central Valley, with the state of Maxson, Sac-Town, and the Redding Territory (presumably also where most of the fighting of the NCR-Brotherhood War is).
The Hub itself, the economic and logistical center of NCR, presumably ties the country together with its roads and rail networks. If you take that, NCR is cut in three, cut off from each other and able to be taken piecemeal without coordination or supplies between the thirds. Once the Hub is taken I presume Caesar will want to go for Shady Sands first, take out the political center and break the will of the south and the valley to fight. The Legion moves up the valley itself, just a straight flat line without many natural defenses. Lightly settled with the Brotherhood pocketed throughout, the Central Valley will crumble pretty easily. It migth fall apart without the Legion even entering it, just from political chaos and the Brotherhood staging breakout raids from their bunkers and hidden outposts.
Surrounded by mountains and compact, Boneyard-Dayglow would be a tougher nut to crack, and it might take a few years. But there's a lot of different entrance points, and now with the captured materiel of at least the Hub and the state of Shady behind it, the Legion will eventually find a weak link and rush in like a flood (assuming, of course, Caesar doesn't die first).
The battle for the Hub itself would probably be pre-saged by a Battle of Barstow. This is where Long 15 and Interstate 40 meet, Interstate 40 being an alternative route for Caesar's armies for Arizona. Control of Barstow would also allow Caesar to block any NCR reinforcements or flanking from the Boneyard/Dayglow, they'd be forced to take the long way through Santa Clarita. And once Barstow's taken, it's just two days march from the Hub itself.
Now, does the Legion win the Battle of Barstow or the Battle of the Hub? I think not. As others mention, the NCR has superior mobility/maneuver to the Legion, not only does it have mechanized units it also has rail infrastructure in its heartland. While the NCR failed to utilize it's mechanized capacity in the eastern Mojave, now that the Legion is advancing into its core territory they'll have to pull their heads out. And since the Legion is now marching into the part of NCR supplied by rail lines, they'll be shocked at how quickly the Californians are able to get men and materiel in place.
During the week's march from the mountain pass to Barstow, I think the NCR will turn the tables on Caesar and harass him the whole way down. If Caesar attempts to campaign through the desert to obviate ambushes and attacks, he'll find himself outmaneuvered by mechanized NCR forces in the wide open high desert. He'll have to stick to Long 15 and Interstate 40, dealing with attacks and needling the whole time. And the NCR will be ready for him at Barstow.