I think it was this episode where ST TOS states they avoided a 3rd World War.
I do believe in TOS (a taste of armageddon) they did mention that they did use nuclear weapons and nearly destroyed themselves.
the ugenics war happened in 1992 and 50 years later the nuclear war happened, and enterprise took place in 2151 so 100 years is more than enough time to rebuild society. Look how far we have come in the last 100 years technologically. go here to see the timeline
http://en.wikipedia.org/wiki/Timeline_of_Star_Trek
My theory is that the two conflicts were interconnected. The Eugenics Wars were fought covertly as most of the wars between the 70's and 90's and WWIII occurred as a result of a mass destabilization of global governments and economies as a result of the revelation of the Eugenics Wars to the public. I mean, even TOS mentioned Philip Green as being one of the villains of WWIII seperate from Khan and the Eugenics Wars. It's kind of like an observer from the future could lump both World Wars as one conflict because the Treaty of Versailles arguably led to WWII