When The Bomb was dropped, it was referred to as "the atomic bomb" (edit: and was referred to as such by Truman announcing its use on Hiroshima). The famous Korean War song refers to "if General MacArthur drops the atomic bomb", and a cursory google search results in finds such as "Atomic Power", a film supposedly created in 1946. However, by the time of the Cuban Missile Crisis, Kennedy spoke of "nuclear warheads" and "nuclear missiles" in his October 22 speech, and these days, the use of "atomic" instead of "nuclear" seems almost archaic. We might maybe use "atomic bomb", but warheads, weapons, or the power from reactors are "nuclear". When did this change in terminology occur, and is there a specific reason behind it, or is it merely a more-or-less-random change in language use?
(I also have to think of Frank Herbert's use of the term "atomics" when he wrote Dune in the early 60ies, which sounds almost deliberately archaic and estranging, but I'm not sure how relevant this really is.)
The preferred term for the scientists used the root of "nucleus" even in the 1940s, but the term "atomic bomb" (which dates from the 1910s) was more popularly known already (through science fiction and speculation), so it was used instead.
The shift towards "nuclear" took place in the context of the nuclear power industry. By the late 1950s "nuclear" had overtaken "atomic," which was starting to sound quaint.
You can see this trend with Google Ngrams very nicely. If you break it into sub-terms, you can see more specifically how "nuclear weapons" and "nuclear power" shift things starting in the mid-1950s.
As for why it changed — I don't think I'd call it random. By the 1950s "atomic" started to sound old-fashioned and, again, was never very scientifically specific (chemical reactions are "atomic" reactions as well — only "nuclear" specifies exactly what part of the atom is reacting with things like fission or fusion).
To my knowledge, there was never any moment in which anyone in the US government (or the US Atomic Energy Commission — "atomic" again!) specifically said, "let's stop using 'atomic,'" but there was a definite shift in the 1950s.
Thank you all for your answers! So, to summarize, what I get from your answers is the following hypothesis: