Tongues of the Moon (1964) - a novel by Philip Jose Farmer. [Expanded from a novelet with same title in Amazing September 1961.]

North America had fallen to the Communists, who found that the pieces were hard to digest. American Communists had become a trial to Moscow, alarming the Russians with their increasing power and influence. America had recovered and was once again a great industrial empire. Ostensibly under Russian control, the Americans were pushing subtly and not-so-subtly to get their own way. But they were a sideshow compared to the Argentinian/South African "Axis." They had warned the Soviets that if ecconomic pressures and discriminations were not stopped, the Axes would explode cobalt bombs - poisoning the entire Earth! Not believing the Soviet's placatory stance, the Axis thought it could win a conventional nuclear war if they struck fast and first. They were wrong. [That's the backstory, as explained in passing.]

After an atomic war has devastated the Earth, there are 2 pockets of humanity left: a few hundred Communists on the Moon, and a few thousand in an Axis base on Mars. Somewhere on Earth is a super-weapon: a planetbuster that could be used to crack Mars. And the Moon survivors plan an expedition to ravaged Earth to get it...