The term "atomic bomb" was first popularized in 1914 by H.G. Wells in his novel The World Set Free. It was the first real attempt to take the emerging science of radioactivity, and its promise of unlocking "atomic energy," and turning it into a sci-fi prediction of the future. (He based his scientific understanding on that gleaned from Frederick Soddy's The Interpretation of Radium, published in 1909.) This set off three decades of speculation about "atomic bombs" in the popular press, well before nuclear fission was discovered.

The point of this is to emphasize that in the summer of 1945, when the US military was secretly making plans to announce the first atomic bombs — the announcements that would follow the first use of them in war — they were cognizant of the fact that people already had a vague idea that a thing called an "atomic bomb" was possible and that this new invention slotted well into that existing idea. The scientists working on the weapon preferred emphasizing the nuclear nature of it, because it is really its nucleus that is responsible for the energy that makes it so important (chemical reactions are also "atomic" in the sense that use electrons, which are part of the atom as well).

So, for example, the internal, official history of the Manhattan Project that was written by Princeton physicist Henry DeWolf Smyth during the war was initially meant to be called NUCLEAR BOMBS. It was feared this would confuse the very public it was meant to inform, so the military recommended changing it to ATOMIC BOMBS. As it was, they were so anxious about the attention-grabbing nature of the title that they didn't actually print it on any of the copies, instead just printing a lengthy subtitle ("A General Account of the Development of Methods of Using Atomic Energy for Military Purposes") with the intent of applying the actual title right before release with a rubber stamp. They forgot to apply it to all copies except for the one filed at the Library of Congress for copyright purposes (here's what it looks like, photo by me), and so instead the subtitle was regarded as the title, and everybody ended up calling it the Smyth Report anyway.

The point being, calling them "atomic bombs" was always regarded as pretty quaint and inaccurate by the scientists, but nobody really cared what the scientists thought. That was the term that people understood them to be, and it became self-sustaining through the 1940s and 1950s.

So what changed? If you look on Google Ngrams, you can see that there is basically a cross-over point in the terminology between "atomic bomb" and "nuclear weapons" around 1953 or so, and a generalized one between "atomic" and "nuclear" around 1956 or so. These are interesting years to contemplate; the beginning of the 1950s saw the invention of the "hydrogen bomb," which is to say a thermonuclear weapon, and so there was quite a lot of discussion contrasting it with the "atomic bomb" which was characterized as being fission-only. Creating a new, overarching term for both types of weapons was probably part of the reason for the switch to a more inclusive terminology, but so also was the growing familiarity with the word "nuclear" in general that was being pushed as part of the development of nuclear power (which was sometimes called "atomic power" for awhile, but that eclipse in terminology happened pretty quickly). Furthermore, by the 1950s, many nuclear weapons were not, in fact, "bombs," in the sense of something that is dropped by a plane. This was the beginning of the missile age, and the creation of a gigantic stockpile of incredible technical variance (missiles of all sorts and ranges, tactical nuclear weapons, nuclear torpedos, nuclear landmines, you name it).

Anyway, it is clear that the official, governmental language of these arms switched to "nuclear weapons" starting in the 1950s, no doubt for a variety of the above reasons, and the popular understanding eventually dragged along with it. This ended up rendering the terms "atomic bomb" and "hydrogen bomb" as feeling specific to the 1940s and early 1950s. So it is not uncommon to use them in these historical references, such as with regards to Hiroshima and Nagasaki and the H-bomb debate and early H-bomb tests, but the terminology is antiquated today, in the sense that nobody "informed" would use it to refer to modern weapons.

My favorite book on nuclear culture and attitudes is Spencer Weart's Nuclear Fear, and he does a good job of characterizing in general how the 1950s saw a major shift in how all things "nuclear" were regarded, as part of the trends mentioned above.