Well (since we are discussing canon):Or, if you are so inclined, you can go the route offered up by Greg Cox's Eugenics Wars books which depicted them as a series of secret wars that the majority of the people of Earth were unaware of the full extent of, or that certain natural and manmade disasters were actually salvos in that war. Certain populations could experience localized "dark ages" as a result of these disasters while bigger countries were able to largely maintain stability.
1) Star Trek books are not considered canon.
2) Even If you accept what's in the books - sorry, but that doesn't juive with Mr. Spock's comments from the actual "Space Seed" episode. It just doesn't:
^^^SPOCK: Your Earth was on the verge of a dark ages. Whole populations were being bombed out of existence. A group of criminals could have been dealt with far more efficiently than wasting one of their most advanced spaceships.
See bolded part
and again from my earlier post:
^^^^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.
see bolded part - sorry but you really want to say that from 1993-1996; all that could be kept secret from the world at large?
No, I'd rather say what happened: Neither Branon Bragga or Joe Menosky cared about TOS or STII:TWoK canon because they felt "Future's End" was a great story; and they wanted to have the crew end up in the real world year the episode was being filmed.
IMO - It's amazing all the mental gymnastics people will go through to claim "Future's End" is consistent with canon; yet, the idea the the U.S.S. Kelvin (and it's Bridge) are NOT in the Prime Timeline, even though technically the Divergence in said Timeline didn't occur until Nero's ship appeared from the singularity in the 2230ies and destroyed the Kelvin and killed Kirk's father.
And yeah, we're all fans here and free to have/interpret canon/fanon, etc; as we like. But IF you accept 'canon' as what appeared on screen (in dialogue and visuals) - my rationalization for the U.S.S Shenzhou Bridge being in the actual 'Prime Timeline' is way simpler, and straightforward then anything VOY fans can come up with to say "Oh no, "Endgame" fits within the 'Prime Timeline' with no problem."

Again, fans are free to interpret Star Trek however they want.