An attempt at a continuity preserving kludge here...
Relevant excerpts from "Space Seed":
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SPOCK: Much older. DY-100 class, to be exact. Captain, the last such vessel was built centuries ago, back in the 1990s.
...
SPOCK: Hull surface is pitted with meteor scars. However, scanners make out a name. SS Botany Bay.
KIRK: Then you can check the registry.
SPOCK: No such vessel listed. Records of that period are fragmentary, however. The mid-1990s was the era of your last so-called World War.
MCCOY: The Eugenics Wars.
SPOCK: Of course. Your attempt to improve the race through selective breeding.
MCCOY: Now, wait a minute. Not our attempt, Mister Spock. A group of ambitious scientists. I'm sure you know the type. Devoted to logic, completely unemotional
...
KIRK: Suspended animation.
MARLA: I've seen old photographs of this. Necessary because of the time involved in space travel until about the year 2018. It took years just to travel from one planet to another.
...
KIRK: Would you estimate him to be a product of selective breeding?
SPOCK: There is that possibility, Captain. His age would be correct. In 1993, a group of these young supermen did seize power simultaneously in over forty nations.
...
KIRK: Name, Khan, as we know him today. (Spock changes the picture) Name, Khan Noonien Singh.
SPOCK: From 1992 through 1996, absolute ruler of more than a quarter of your world. From Asia through the Middle East.
MCCOY: The last of the tyrants to be overthrown.
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Put all that together, and it seems very clear that Khan and the Eugenics Wars date to the 1990s.
Yet it's also true that from TNG forward (most notably in "Encounter at Farpoint" and
First Contact), Trek continuity has situated World War III in the mid-21st century.
Solution: note the dialogue I bolded above. Spock says "your last so-called World War"... and McCoy
corrects him, having caught him in a rare mistake, by giving the correct name of the conflict. Watch the original; listen to the line delivery; it's a perfectly plausible interpretation.