The question is because of an /r/askreddit thread -
I couldn't find any information of any program or guidelines for such a thing.
The only in-depth discussion of these tattoos that I have seen is this one from CONELRAD Adjacent — the practice did exist but seems to have been relatively localized to only a few places in the US (not a national program) and not all that common. Still, it is a pretty interesting little bit of history. CONELRAD Adjacent in general is a great source for all of these kinds of Civil Defense minutiae/trivia; nobody else really works on it in my experience.