From TOS we can surmise the following:
- There was a third World War, in which 37 million died.
- The Western Bloc / Eastern Bloc Cold War of the latter twentieth century did not result in a large-scale hot war, as many people feared might happen in the real world.
- Instead, World War III took place in the 1990s and involved power struggles between genetic supermen, with "whole populations bombed out of existence." It was also known as "the Eugenics Wars."
All that goes together just fine.
But later, in TNG, World War III was retroactively moved forward into the 21st century, as shown in "Encounter at Farpoint" and "Star Trek: First Contact." WWIII did not seem to be related to the Eugenics Wars anymore, and the death toll was now 600 million instead of 37 million.
Kor