In 1962 the band of mutants who rose up to oppose Shaw never stopped the Americans and Soviets bombing each other. Erik and the others were too late in arriving and consequently World War III was unleashed upon the world and the reveal of the mutants soon followed. Sebastian Shaw's dream, the rise of the children of the atom, was now a reality. Erik and Charles formed the X men together and are an underground resistance, working to salvage the world from Shaw and his army.
Atom is set in 1964, in a post apocalypse setting following real time. The game is canon compliant up until the point where the X Men stopped the Soviets from crossing into Cuba's waters and has other Marvel themes in play.