If you mean totally hypothetically, the British novelist H.G. Wells took ideas about radioactivity and the energy potential of "the atom" and put them into his science fiction book, The World Set Free (1914). These were not fission bombs, but weapons that released a sort of never-quenched destructive fire. In the frontispiece of the book, he acknowledged the chemist Frederick Soddy's work of popular science, The Interpretation of Radium (1909) as giving him the idea, though Soddy does not describe weapons in his book on radioactivity and its potential (though he did initiate many of the tropes that became quite common in explaining how much energy was "locked away" in the atom). Wells' work touched off considerable speculation about the possibility of "atomic bombs" writ large, and there were many articles, speeches (including one by a young Winston Churchill), and books written that brought up the idea that if scientists could make radioactivity work at their will, an intense explosive could be developed.

Jumping ahead: Leo Szilard, a Hungarian physicist who had fled the rise of the Nazis in Europe was in London in 1933. While there, he read a report that said that Ernest Rutherford (the major British physicist at the time, who worked with Soddy on "transmutation" of atoms) had disparaged popular talk of atomic bombs and industrial uses of atomic energy as "moonshine," which is to say, nonsense. Szilard took offense at this, thinking Rutherford was being insufficiently appreciative of the possibilities. He also, it should be said, was a deep reader of H.G. Wells and found his visions of the future startlingly possible. Szilard set his mind to the task of figuring out a way to make a nuclear reactor and/or explosive.

Szilard came up with the idea of neutron-induced nuclear chain reaction. The idea was this: neutrons, a subatomic particle that had only been discovered the year prior, lacked an electrical charge (unlike the positive protons or negative electrons). That meant that they were not repelled by atoms, and could penetrate deep into them. (This was not a unique insight of Szilard's; this fact made neutrons a "hot" topic among physicists.) Szilard knew there were types of reactions in an element would absorb one radioactive particle and as a result eject another (or several). For example, if you bombard beryllium with alpha particles, it will release neutrons.

Szilard's idea, as he later put it, was thus:

...if we could find an element which is split by neutrons and which would emit two neutrons when it absorbs one neutron, such an element, if assembled in sufficiently large mass, could sustain a nuclear chain reaction.

Did such an element exist? Not that he knew of. Nature didn't prohibit it, but nobody had found one, and nobody had set out to look for one. Szilard attempted to secure some funds to look into this, but they were lacking. Nobody shared his interest (Rutherford literally threw him out of his office for annoying him, and because Szilard attempted to patent this idea despite it being underdeveloped), and he himself was not a particularly inspired or talented experimentalist (he was an idea man).

Szilard attempted to patent the idea, as a way of getting official attention for it, and to keep the patent secret. But, again, he really didn't know how this would work. It was all based on a hypothetical nuclear reaction, and he didn't know what that might be. If you read the patent today with a technical eye, you find it is a mess of good ideas and bad ideas. Separating out the good from the bad would require the passage of time, which revealed the "right" answers. Szilard ended up putting the idea aside for many years, but it was in the back of his head.

In late 1938, a team of chemists and physicists working out of Berlin under the supervision of radiochemist Otto Hahn were attempting to debug a complicated experiment that the had won famed Italian physicist Enrico Fermi the Nobel Prize that same year. Fermi had bombarded uranium with low-energy ("slow") neutrons, and found that it had produced unexpected radioactivity as a result. Fermi had proclaimed that he had created new, "transuranic" (heavier than uranium) elements for the first time.

Hahn and his collaborators doubted that this was what had happened, and were repeating the same work but with the careful chemical attention that Hahn could provide. They bombarded the uranium with the slow-neutrons (to slow a neutron, you first send it through a hydrogen-rich medium, like paraffin). Then Hahn used all of his clever radiochemical techniques to figure out what was remaining: what is uranium, or something else? And what was the something else, chemically? And were the radioactive signatures of the "something else" recognizable?

What he found was that one of the main by-products of the reaction was a radioactive form of barium. This was very unexpected, since barium (atomic number 56) is very different from uranium (atomic number 92). Most nuclear reactions involve slight changes to the atomic number or atomic weight (jumps of 1 or 2 in one direction or the other), and having something so small result was puzzling.

He sent these results in letter form to one of his collaborators, the physicist Lise Meitner, for analysis. Meitner was an Austrian who had worked for years in Hahn's laboratory, and despite being Jewish she had managed to wait until a very late date to flee the country's Nazi rule. She was at that point in Sweden, as a refugee scientist. Her nephew, the physicist Otto Frisch, was also with her. Meitner and Frisch discussed the results, and concluded that the only reasonable interpretation was that the uranium atom had been cleaved roughly in half. Frisch called this phenomena "fission," in an allusion to the biological splitting of a cell. Meitner also immediately recognized that this would release a lot of energy. She wrote this back to Hahn, and both Hahn and Meitner prepared papers announcing their discovery.

Nuclear fission by itself was not an atomic bomb. The energy released per reaction is large for an atom but small for a human being. It is about as much energy as is needed to disturb a single speck of dust. That's still pretty impressive (from a size perspective, an atom disturbing a speck of dust is like an ant being able to wobble a Space Shuttle; it is roughly 200 million times more energy than is released through any individual chemical reactions), but a speck of dust is a speck of dust. No one involved in the experiment in Berlin seems to have considered it of any military importance.

But Szilard was still out there, now at Columbia University in New York, and his mind was "primed" to think about any neutron-based reaction as the possible catalyst for realizing his chain reaction. When he heard, through word of mouth (the news traveled relatively fast in the physics community, even prior to publication) in early 1939 about the Hahn-Meitner work, he immediately though back to his chain reaction idea. Here was a potential candidate reaction: the thing he needed to know was, did the splitting of uranium also produce neutrons? He began a search at Columbia for these "secondary neutrons," that would confirm whether an atomic bomb might be physically possible.

He was not the only one to begin thinking along these lines after the discovery of fission. Scientists in France, the US West Coast, and Germany also began thinking along these lines. Szilard attempted to encourage scientists outside of Axis states not to publish on these matters, but essentially failed. Soon the fact that secondary neutrons did exist, which Szilard confirmed himself, became public knowledge, and everyone was thinking of atomic bombs.

Szilard later described the experiment he set up in the search for secondary neutrons:

[On March 3, 1939] everything was ready and all we had to do was to turn a switch, lean back, and watch the screen of a television tube. If flashes of light appeared on the screen, that would mean that neutrons were emitted in the fission process of uranium and that this in turn would mean that the large-scale liberation of atomic energy was just around the corner. We turned the switch and we saw the flashes. We watched them for a little while and then switched everything off and went home. That night there was very little doubt in my mind that the world was heading for grief.

Anyway. I have written this all out so you can see that the question of "first realized" is a tricky one. Szilard is probably the best candidate for a single person. But as with many complex phenomena, there are many people who play a role. Even the description I have given above is heavily abbreviated, leaving out contributions from, say, Niels Bohr, Ida Noddack, Frederic and Irene Joliot-Curie, and many others, to say nothing of the work that was required to turn this idea into something truly plausible, much less real. For the whole long story, Rhodes' The Making of the Atomic Bomb (1986) is still a good work, especially for the scientific background (which takes up about half of the book).