The short answer is "no".
How a Theory is Born
A good analogy to the relationship you speculate on between Wells and the atom bomb is the influences on Charles Darwin. Before Darwin ever dreamed of the theory that would later make him famous, and even before he was born, works on evolution had already appeared in print, most famously the works of Jean-Baptiste Lamarck. But there was actually an evolutionist in Darwin's own family tree. His grandfather, the British polymath Erasmus Darwin, had published a book of science-based poetry called Zoönomia, or the Laws of Organic Life, in 1794. Zoönomia contained speculation about how animal and plant life changes over time, and in retrospect, it bears some minor similarities to the modern theory of evolution. Charles Darwin was familiar with both of these works, but he seems to have rejected them from as early as he began to study science.
He came to see that a form of evolutionary theory was indeed correct, of course, but he did so on the basis of an extraordinary amount of evidence, some of which he had discovered himself on his famous voyage aboard the HMS Beagle. When he published his iconic work, On the Origin of Species, it was not a result of his familiarity with previous variations on the theme of evolution; it was based on a tremendous amount of research, including his own original findings, and was distinct from earlier forms of vaguely similar ideas in the sense that he followed the scientific method very scrupulously. For the first time, evolution was described in a rigidly scientific manner, relying almost exclusively on actual evidence rather than wild speculation.
He wasn't the first person to theorize that species changed over time; he was the first to attribute this phenomenon to natural selection. He remained highly critical (even contemptuous) of Lamarck, and while he presumably felt some degree of affection for Erasmus Darwin, who was, after all, his grandfather, he didn't owe anything to Erasmus' work (which was more artistic than scientific in tone and origin). Neither Lamarck nor Erasmus were the inspiration for Darwin's discovery - Darwin himself was the man responsible for the breakthrough.
This is the way science works. A person might have a sort of epiphany after reading a fictional story, but it would be meaningless if the person in question didn't have some prior experience in the relevant field of study. If the field of study already exists, it is because we already know it is a feasible source of new information. You can't attribute discoveries within that field to the fact that someone working in the field read a work of fiction, no matter how interesting or appealing the idea that the fictional story led to the discovery may be.
The Advent of the Atomic Age
We generally associate the development of nuclear energy and weapons to the theory of special relativity, which Einstein put forth in the first decade of the twentieth century (and improved upon in his theory of general relativity years later). The theory of special relativity was published more than a decade before the publication of Wells' book. Many other people were involved in the process, of course, including Leó Szilárd, Enrico Fermi (thanks to Praxis for pointing out Fermi's relevance), Marie Curie, and many others. Yes, Szilárd had read The World Set Free a year or two before he conceived of the idea of neutron chain reactions, but this isn't as important a connection as it may first appear. He had already been a physicist for over 13 years by the time he patented the concept of chain reactions, and 11 years before he read The World Set Free. Clearly, his lengthy experience in the field was of more import than what he did in his spare time, including reading the book.
In fact, Szilárd himself attributed his idea to the outrage he felt at hearing a speech by Ernest Rutherford. In the speech, Rutherford said:
"We might in these processes obtain very much more energy than the proton supplied, but on the average we could not expect to obtain energy in this way. It was a very poor and inefficient way of producing energy, and anyone who looked for a source of power in the transformation of the atoms was talking moonshine. But the subject was scientifically interesting because it gave insight into the atoms."
The Wikipedia page (already linked and) for Szilárd explains:
Szilard was so annoyed at Rutherford's dismissal that he conceived of the idea of nuclear chain reaction (analogous to a chemical chain reaction), using recently discovered neutrons. The idea did not use the mechanism of nuclear fission, which was not yet discovered, but Szilard realized that if neutrons could initiate any sort of energy-producing nuclear reaction, such as the one that had occurred in lithium, and could be produced themselves by the same reaction, energy might be obtained with little input, since the reaction would be self-sustaining. The following year he filed for a patent on the concept of the neutron-induced nuclear chain reaction. Richard Rhodes described Szilard's moment of inspiration:
"In London, where Southampton Row passes Russell Square, across from the British Museum in Bloomsbury, Leo Szilard waited irritably one gray Depression morning for the stoplight to change. A trace of rain had fallen during the night; Tuesday, September 12, 1933, dawned cool, humid and dull. Drizzling rain would begin again in early afternoon. When Szilard told the story later he never mentioned his destination that morning. He may have had none; he often walked to think. In any case another destination intervened. The stoplight changed to green. Szilard stepped off the curb. As he crossed the street time cracked open before him and he saw a way to the future, death into the world and all our woes, the shape of things to come."
An article from Scientific American continues:
What Szilard realised as he stepped off that curb was that if we found an element that when bombarded by one neutron would release two neutrons, it could lead to a chain reaction that could possibly release vast amounts of energy. Leo Szilárd had discovered the nuclear chain reaction long before anyone else, six years before the discovery of nuclear fission and any inkling that anyone could have had about the release of atomic energy, let alone the woeful apocalyptic future that would await the world because of its release.
Furthermore, the idea of the atom had been around for millennia prior to Wells' story. While the relationship between Wells' book and the atomic bomb is merely coincidental, the fact that Wells' book was published just before a series of fast paced developments in virtually every field of scientific research, including nuclear science, is not coincidental. We were working on the idea because we were finally able to do so, and Wells wrote about it because we were starting to work on it. He says so himself in the first chapter of the book:
The problem which was already being mooted by such scientific men as [William] Ramsay, [Ernest] Rutherford, and [Frederick] Soddy, in the very beginning of the twentieth century, the problem of inducing radio-activity in the heavier elements and so tapping the internal energy of atoms, was solved by a wonderful combination of induction, intuition, and luck by Holsten so soon as the year 1933.
The "Holsten" he mentions is one of the characters in the story. Ramsay, Rutherford (for whom Rutherfordium is named), and Soddy were some of the physicists who pioneered early research into the atom in real life. Ernest Rutherford coined the phrase "atomic energy" in 1903. Wells' knowledge of the atom came from these men - quite the opposite of the relationship you have suggested. Wells only thought of the idea because people like these were already hard at work on the problem of how to harness the power of the atom, not the other way around.
We were starting to work on understanding and harnessing the atom because, although the first mention of the atom occurred thousands of years ago, the atom as we know it was discovered in 1895. Basically, Wells was one of the first authors to mention atomic weapons in a fictional story because the existence of the atom was only confirmed 18 years earlier.
Wells' Atomic Bomb:
If Wells' story was influential enough to make Szilárd think up neutron chain reactions, we might be forgiven for wondering why he bothered to do it at all. Wells' bombs are pretty unimpressive.
Wells's "atomic bombs" have no more force than ordinary high explosive and are rather primitive devices detonated by a "bomb-thrower" biting off "a little celluloid stud."
In case you weren't wondering, the "bomb-thrower" isn't a machine, it's a guy who literally throws atomic bombs like oversized hand grenades:
The gaunt face hardened to grimness, and with both hands the bomb-thrower lifted the big atomic bomb from the box and steadied it against the side. It was a black sphere two feet in diameter. Between its handles was a little celluloid stud, and to this he bent his head until his lips touched it. Then he had to bite in order to let the air in upon the inducive.
This is a tactical weapon, not a strategic weapon. Our own brief experimentation with tactical nuclear weapons for use by frontline soldiers, such as the atomic artillery guns, didn't last very long. It also sounds more like a dirty bomb than a true atomic bomb.
Renowned physicist Lawrence Krauss recently commented on the question of Wells' depiction of the bomb:
"The novel was published in 1914 and anticipated the development of atomic weapons that would be used in war. Decades before they became a harsh reality in the modern world — and perhaps influencing some of the scientists who created the real weapons — the novel coined the term "atomic bombs.".
"Nevertheless not only did Wells' continually burning atomic weapons bear no resemblance to the engines of destruction in the real world," Krauss emphasized, "he thought it would unite the world into one society whereas we are painfully aware that it hasn't changed human thinking, except to divide the world into nuclear haves and have-nots."
Why Build an Atom Bomb?
Once we knew that atoms existed, and that they contained an enormous amount of energy, it was pretty much inevitable that we would try to find a way to use them to kill people. This is, tragically, par for the course. Many of our most important advances in technology are made because we love killing each other so much. We have metals like steel and iron in large part because we wanted to stab everyone with them. We have explosives because Chinese people liked fireworks, but as soon as gunpowder came to the Western world, we used it to lob big balls at each other from cannons. Long before the germ theory of disease was proposed, Mongols were lobbing the corpses of people killed by the plague over city walls in order to spread the disease and wipe out everyone inside, and centuries later, Europeans were deliberately spreading smallpox among the native inhabitants of the Americas.
And more to the point, we have atomic weapons because they are REALLY good at killing a horrendous number of people at once, not because H.G. Wells wrote a book. The World Set Free was published more than a decade after the atom was discovered, and research was already well under way by then. If Wells' book had never been written, we would have built atomic weapons for use in WWII anyway. Once mankind finds a new toy, it is only a matter of time before we figure out how to murder each other with it. Wells was probably the first author to mention atomic weapons, but only because we were rapidly approaching the realization of the hideous potential of such weapons. And if you read the story yourself, you will see that he had no idea how a real atomic bomb might work, or how atomic energy might be harnessed.
In short, no, we didn't invent atomic weapons because of Wells' story. We invented them because we are monsters.
The Birth of Modern Physics
Chronology of the Atomic Age
History of Nuclear Power
History of Nuclear Energy
History of Nuclear Weapons
Updated to address the issue you added to the question:
Nothing in Wells' story would have suggested that the atomic bomb he describes would be capable of destroying entire cities. As my answer already stated, Wells' atomic bomb was no more powerful than conventional weapons, and was only capable of destroying individual buildings. It differed from conventional explosives in that it also spread radiation, making it sound somewhat similar to a so-called "dirty bomb". In the citations directly above, you will find that the first well known person to seriously and publicly discuss the possibility of atomic weapons being able to destroy large areas was possibly Winston Churchill.