Well I've always associated WWIII and the Eugenic Wars. In Encounter at Farpoint and ST:FC it's clear the former is now in the 21st century, whereas I think it was implied to be in the 20th in TOS.
I've always considered the Eugenics Wars to be a series of separate but related wars lasting on and off from ~1992-2079 that were thematically connected by later historians, much like the Hundred Years' War was in reality.
That way, if you want you can encompass any or all of the following into these separate but related Eugenics Wars:
- Khan's and the other Augments Eugenics Wars of 1992-96
-the confusion over the death toll in WWIII between Spock's, Riker's, Phlox's and other estimates (30/35/37/or 600 million dead), and also the confusion between whether Khan's war was WWIII or whether it was a separate conflict
- the neo-Trotskyist revolts in Europe (though that seems more related to the global economic collapse of the time)
- the nuclear exchange in WWIII in 2053 with the Eastern Coalition that killed over 600 million people
- Colonel Green's earlier eco-terrorist conflicts and mass genocide in WWIII in 2026 that killed 37 million people
- the post-atomic horror lasting till at least after 2079
- the wars in the 21st century mentioned in "The Savage Curtain"
- the genetic engineering of Dr. Stavos Keniclius well into the 21st century before it was finally banned and he was exiled like Khan
- the still recovering from WWIII Earth in the 22nd century from "Up the Long Ladder"
- Archer's great-grandfather fighting in the Eugenics Wars in North Africa, which would have to be long after 1992-96
Hell, if you really wanted to stretch it, you could call the events of the Augments and Klingon arc in Enterprise S4 the last gasp of the Eugenics Wars, which could explain Admiral Bennett in DS9 thinking they extended into the 22nd century.