So? What makes you think WWIII would be entirely nuclear? For all we know, only India and Pakistan may have exchanged nuclear fire in their theatre of WWIII. No reason to think the entire globe was engulfed in a nuclear conflict. Certainly the 2060s don't look like a post-apocalyptic wasteland, since Cochrane was able to build and launch a warp-capable ship! Think of the resources behind that effort - how could it have happened if the entire world was reduced to the levels you assume?
For the record, we get a glimpse at the start of World War III in The Lost Era: The Sundered
, by Andy Mangels and Michael A. Martin. The Sundered establishes that the nuclear strikes commence on 1 May 2053. It specifically establishes that London, New York City, Tel Aviv, Jerusalem, Mecca, Riyadh, Karachi, and New Delhi were hit by atmospheric detonations, and that nuclear detonations happened on every inhabited continent. The novel implies that one hundred million likely died in the immediate attacks on the first day, and establishes that the planetary economy collapses as a result of the war. It also establishes that the self-sufficient O'Neil colonies established on a number of hollowed-out asteroids in Earth orbit helped Zefram Cochrane obtain the materials needed to launch the Phoenix
, which is implied to be an independent continuation of a U.S. government project Cochrane had led before the government collapsed.