When was the nuclear bomb invented?

H. G. Wells' novel The World Set Free published in 1914 was the first usage of the term atomic bomb. He reasonably correctly estimated their energy release (based on Einstein's special relativity) but could not imagine that the energy could be released faster than was possible in chemical bombs, so his atomic bomb explosions took days to weeks instead of microseconds.

Leo Szilard invented the neutron chain reaction, which makes both atomic bombs and nuclear reactors possible in 1933 while living in London. He patented it in 1934 and when the patent (GB630726) was granted in 1936 he sold it to the British Admiralty who classified it to keep the information from the Nazis. The only problem was that no material was known at that time that would support a neutron chain reaction.

In 1939 a joint German and Swedish team in repeating an experiment performed originally in 1938 by Enrico Fermi (who had mistakenly decided that when he bombarded uranium with neutrons that he had made transuranic elements) determined that neutrons split the uranium atom into two smaller intensely radioactive atoms plus several neutrons. It was not necessary to see Szilard's classified patent to realize that uranium was the material that would support a neutron chain reaction. It took a little more research to identify that only the rare uranium-235 isotope (0.7%) would support a neutron chain reaction, not the plentiful uranium-238 isotope (99.3%).

The British did not declassify Szilard's patent until 1949, after the Soviet atomic bomb test.