I imagine it would depend. Some of the warlord states might get outside support from NATO just because, well, they aren't communist. Even discounting that a few of them are democratic, market liberal states, or trying to be, well, it's not like we've never propped up less than pleasant regimes as long as they were opposed to communism.
Nukes shouldn't really be a concern. The warlord states that reach a position to even attempt to acquire nukes don't do so until the 70s, and I think it takes until the 80s for their nuclear programs to reach a point in which they're even able to produce a single bomb. (And that's after one of them has basically unified the parts of Russia not controlled by the Nazis or their puppets.)
That said, the warlord states are still going to be in a really shitty condition. The parts of the Soviet Union that didn't get swapped will probably be able to conquer all of the warlord states. Even if the rest of the Warsaw Pact takes advantage of things to tell the Soviets to fuck off.
Of course, the Soviets will be ruined by having to retake so much of their land and rebuild it all from the absolutely squalid and economically ruinous condition it's all in. So the Cold War is basically done with much sooner. Which honestly might result in the West becoming friendlier to leftism for no other reason than because you have 30 less years of the Soviet Union being the great big enemy.
Because, honestly, without half a century of a sustained Red Scare do you really think a fear of communism and socialism is going to last long enough for it to have had the impact it has had in our timeline?