I don't think it comes from the Cold War period at all (and thus falls out of our 20-year restriction on this subreddit). I have read a lot of Civil Defense sorts of documents, including popular writings (which sometimes do include erroneous information), and have never seen it suggested. It is, in some ways, a very strange sort of suggestion — the geometry of it doesn't hold up at all, to say nothing of its ignorance of nuclear effects (they do not scale linearly; the mushroom clouds do not have a single fixed "height" except in still photographs; you can still be exposed to dangerous downwind fallout even if you can't see the mushroom cloud or blast, especially for high-yield weapons).

My guess is that it is just an Internet rumor, one of those "you might think this image is straightforward, but there is a secret mystery to it!" sorts of things. It has zero factual basis. I am not sure it predates the release of Fallout 3 in 2008, which seems to have been the first to use the thumbs-up imagery. I have not seen anything to imply that the creators of the Fallout series ever intended the gesture to be anything other than an expression of faux optimism in the face of a crushing post-apocalyptic reality.