I would suspect not much more than India and Pakistan, with interests or occupation forces perhaps in the unstable, Augment-hating but still oil-rich Islamic Middle East. Countries with major Augment populations in prominent political positions were probably ruled indirectly, or even through an official supranational body, by a cosmopolitan society of thousands of political elites, representing tens of millions Augments, whose interests were not those of nationality. Khan Noonien Singh would have been at or near the apex of this pile of superhumans.
They probably had very close ties to Augment communities throughout the world, and attempts to defend those communities against increasing genetic-lamer animosity, aggression, and persecution, particularly in the religious, anti-genegineering United States, is my idea for the trigger that led to nuclear war.
The U.S. might have hit key sites (with nuclear weapons?) with the goal of killing as many Augments as possible, and destroying the Augies' ability to coordinate and resist, instead of taking out the enemy nations as a whole, counting on the possibly-oppressed normo population to welcome liberation. (We always expect that, and we are often disappointed.

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I suppose the lingering resentment toward the collateral damage caused by Americans (and American nukes?) might have been a root cause of World War III, obviating the need to consoldiate the two wars.
Somebody had to bomb those whole populations out of existence, and it wasn't Khan who started it, unless Kirk is a complete jackass.
"We found Hitler's brain? Well, let's get him fixed up with an android body or something--he was, after all, the best of Nazis."