In literature and other media from the 1940s and 1950s, I often see the terms "atomic energy" or "atomic warfare" which sound mildly archaic in American English. By the 1960s and 1970s, it seems like "nuclear energy" or "nuclear warfare" had become the preferred terms for these concepts.
You can use Google Ngrams to get a good sense of this sort of thing. If you break it down further, it can reveal more specific trends about what terms changed. But the short answer is, "nuclear" first crept into the popular parlance through "nuclear power" and then "nuclear weapons." The scientists preferred "nuclear" as early as even 1945 ("atomic" in their mind refers to the entire atom, including the electrons, and not specifically the nucleus), but it was not a term that was considered popularly well-understood enough by the government/military.