So? What makes you think WWIII would be entirely nuclear?
It seems inevitable that it would be.
And in any case, no matter how unlikely, isn't it safer to ASSUME that any WW III will be entirely nuclear? Paranoid, maybe; but it'd make it just that less likely to actually occur. If more people assume the worst, they may not risk actually instigating it.
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?
We actually have no idea what the rest of the world looked like. Literally all we ever saw was a few acres of forest and a missile silo. For all we know, entire cities for two hundred miles in every direction were ashes.