King Daniel wrote:
It was also said or implied somewhen in TOS that Earth avoided nuclear war althogether. One of the old Best of Trek books offered a rather clever solution - that the "third world war" mentioned wasn't actually the kind of apocalyptic WWIII we all imagine, but a war between Earth's third world countries.
To be fair, Kirk's line was:
"We faced a similar crisis in our early nuclear age. We found the wisdom not to destroy ourselves."
That doesn't mean there was never any nuclear war on earth, just that whatever incidents may have happened - mankind managed not to destry itself, or the biosphere of the planet.