"World War III itself ultimately lasted from 2026 through 2053, and resulted in the death of some 600 million Humans. By that time, many of the planet's major cities and governments had been destroyed." (http://en.memory-alpha.org/wiki/World_War_III
I have my doubts about WW3 starting in 2026. We see the year 2032 in VOY "One Small Step", and the Charybdis launched in 2037 according to TNG "The Royale", and if they were able to fund big space missions like those, I don't think there could have been a full-on global war going on.