This kind of thing is actually in my wheelhouse (former USAF ICBM Operations officer). A lot of people think history is somehow clouded in mystery, but it's not. Two good books on the topic are Rhodes's
The Making of the Atomic Bomb and
The Nuclear Express by Thomas Reed.
Rhodes's book is exhaustingly detailed about the program. Most people don't realize that nuclear fission of uranium was not some mystery, scientists all over the world worked together and compared notes via scientific journals all the way up through the '30s. Once they realized that an
independently sustained fission event could lead to a weapon of immense power, the race was on.
Germany, led by Werner Heisenberg, pursued a heavy water path that may have been successful had they not been disrupted (twice) by a combination of British and Norwegian commandos, see
https://www.nationalgeographic.com/n...b-heavy-water/ or watch
The Heavy Water War for a dramatization.
The British had the Tube Alloys program, which eventually started working alongside the American Manhattan Project.
The Japanese were also interested but made a political gamble to not pursue it. They, wrongly as it turned out, believed that the Americans would not have the industrial infrastructure to pursue such a program while also fighting a war on two fronts.
The Soviets were also pursuing it, but relied heavily on espionage to get there.
As far as the two bombs deployed, it was a hedge by using two competing designs of a "gun-type" assembly and an "implosion-type." The first used Uranium-235 refined through gas diffusion, a process perfected by the US during the Manhattan Project (at great expense). The gun-type was a much more inefficient design, but the engineering was so simple that we never actually tested it before dropping it on Hiroshima.
The second type, implosion, used Plutonium-239 produced via reactors at Oak Ridge, Tennessee (the experimental "X" reactor used to test the production theory) and the full-scale reactors at Hanford, Washington. The engineering was so complicated that many of the engineers thought it was impossible to do correctly, right up until the successful Trinity test.
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