When and why did people drop the word "atomic" in terms such as "atomic power", "atomic weapons" etc, and replace "atomic" with "nuclear"?

"Nuclear" specifically refers to reactions that involve the nucleus; "atomic" is vaguer because it refers to the entire atom, which includes electrons. So if you are being pedantic you can say a chemical reaction as an "atomic reaction" because it involves electrons being exchanged. Whereas a "nuclear reaction" will only involve changes to the nucleus.

This was known in the 1940s and the scientists who developed the atomic bomb preferred the term "nuclear" for the above reasons (and because physicists are pedantic people by inclination). However it was felt by the representatives of the US government that any publicity they did about the bomb ought to use the term "atomic" because "nuclear" was a less-well-known word, whereas "atomic" was pretty common. And the concept of "atomic bombs" predated Hiroshima considerably as an item in fiction and popular science. So the first articles and books about the bomb all used the terms "atomic bomb" and "atomic power" and "atomic energy," because these were already part of the popular English lexicon, despite their being somewhat inspecific.

We can use something like Google Ngrams to pinpoint the shift. Here are just "atomic" and "nuclear", and you can see that both were used, but "atomic" slightly edged out "nuclear" until a big switch around 1955. If we narrow it down to technological categories, like so, we can have a bit more confidence in the results (since we aren't going to get unrelated uses of either adjective), and once again 1955 seems to be the magic year.

What's going on in 1955? Several things, but the major trends I'd note are:

The development of "thermonuclear" weapons. This by itself probably switched things around a bit, because the category of "atomic bomb" tended to be used only for fission weapons, with "hydrogen bomb" for thermonuclear weapons, but broadly speaking the entire class of weapons would be known as "nuclear weapons" starting in the late 1950s/early 1960s. (To clarify since someone else asked: thermonuclear means you have achieved the conditions necessary for nuclear fusion reactions to take place. It is usually only used for weapons that derive some large percentage of their energy output from fusion. All nuclear weapons also derive energy from fission, and even thermonuclear weapons can have well over 50% of their energy from fission, just to complicate things.)
The development and commercialization of nuclear power, which was a result of both the Atoms for Peace initiatives, the Atomic Energy Act of 1954 (which allowed for privatization of nuclear power research and the declassification of many formerly secret concepts), and so on. The "rebranding" of "atomic power" as "nuclear power" seems to have been about making it sound more modern, from what I can tell.

Of course, 1955 just happens to be when they cross over, and as you can see it is not an immediate shift in usage. But basically the Eisenhower era seems to be when there was a concerted effort to change the terminology, and it corresponds with the events I mentioned above fairly well.

The shift seems to have been a conscious effort to reflect on the situation from the late 1950s onward in which "atomic power" and "atomic bombs" was associated with the somewhat crude, early generation of technology from the 1940s, but "nuclear power" and "nuclear weapons" would be used as the more specific, less-old-fashioned sounding term moving forward. Obviously it was not a hegemonic victory — we still talk of atomic bombs, though we tend not to be referring to modern technology when we do so.