1964: Johnson vs Nixon

[President Mitt]

It would be a fairly decent sized LBJ landslide. Richard Nixon was still damaged goods after losing in 1962. Nixon would have been a better campaigner, but I doubt the South would have supported him. Some Southern third party probably grabs the deep South. He would have done better than Goldwater, but not by much.

Nixon wouldn't have pulled to the right like he did in 1968, because there was no backlash against Liberalism in 1964, rather, the opposite. So there's no chance he picks Goldwater, he probably chooses some bland Liberal Republican in the Northeast that really gives him no real advantage.

[President Mitt]

I don't understand why everybody is giving the Deep South to Nixon. They rejected Nixon in 1968 even when Nixon pulled to the Right. In 1964, there was no backlash against New Deal Liberalism, in actuality, it was practically the opposite, with New Deal Liberalism at the peak of political power, so you won't be seeing Nixon pushing to the right. Richard Nixon would probably pull on the mask of the 'Me Too' Republican, claiming he could run the Great Society more efficiently than Johnson. Obviously, this isn't what the South wants. While it is true that the South favored the Republicans over the Democrats by this point (at the presidential level), the South is not prone to lining up behind the "lesser of two evils" as they saw it.

So some Southern Third Party runs and grabs the Deep South. Nixon is really only able to hold on to the West by running an efficient campaign, and is practically shut out everywhere else. Johnson landslide. Honestly though, a Nixon vs LBJ campaign would probably have been one of the dirtiest political campaigns ever.