In history of science, technology and ideas, simultaneous discovery and innovation is the norm, not the exception. Almost every technology (and many ideas) have been invented or discovered very nearly simultaneously by multiple inventors once the intellectual and physical "building blocks" required for the invention have become available. These building blocks include but are not limited to knowledge, other ideas, other technologies, components, and sometimes even the zeitgeist of the times.

For almost any given technological field, there will be a relatively worldwide distribution of interested parties working more or less in the technological frontier. They all have relatively similar knowledge levels and usually are fairly well aware of what the others are doing: science and technology are, generally speaking, global endeavours. Once the aforementioned building blocks - what Brian Arthur [1] calls "components" - become available, they are generally available for all or at least many interested parties at the same time. While the precise timing of inventions is somewhat random (i.e., who gets the priority for a patent), almost every important idea has been discovered independently at almost the same time the person or the group the streamlined popular history says is "the" inventor has invented it. Most often, there has been no real information leaks involved, and even when there have been (as in the case of nuclear weapons), their effects are debatable.

Two examples that come to mind are the invention of the telephone and the dispute between the producers of the movies Deep Impact and Armageddon. Reportedly, the essential telephone patents by Alexander Graham Bell and Elisha Gray were filed within hours of each other, and despite long-running legal battle and controversy, the evidence of either one copying the other's design remains weak. The essential components and knowledge required to construct the telephone were available, and both parties knew how to make use of them.

Similarly, the producers of Deep Impact and Armageddon - both movies about Earth being threatened by a comet or an asteroid, and released at about the same time - allegedly engaged in a costly litigation after accusing each other of idea theft, but apparently, no such charges were proven against either. The idea of cosmic impacts threatening the Earth was part of the zeitgeist when the movie production started: the ideas were sort of floating around at the time and much discussed in the public, and it's not surprising to see several screenwriters to get broadly similar ideas at the same time. Similar examples of simultaneous discovery have been collected for decades; for a relatively recent overview, see [2].

The case with nuclear bombs is most probably similar. During the late 1930s, world-class physicists around the world learned very well that chain reaction of uranium was a possibility and that it could release significant quantities of energy. Using uranium in a bomb was therefore an obvious idea to pretty much anyone who had been following nuclear physics up to the outbreak of the war. Even the Finnish Army asked one Finnish physicist to work out whether the uranium bomb might work (he reported that it would) and the Japanese had a sufficiently advanced bomb program so that they could identify the weapons that had devastated Hiroshima and Nagasaki. What all the other countries aside from United States lacked were the nearly limitless resources that the US could pour into the Manhattan Project to make the idea into reality. Post-war, the Soviet Union of course followed suit, having recognised that the post-war world order would be defined by those with access to nuclear weapons.

Sources

[1] Arthur, B. W. (2009). The Nature of Technology: What it is and how it evolves. New York: Free Press.

[2] Lemley, M. A. (2012). The Myth of the Sole Inventor. Michigan Law Review, 110(5), 709–760. https://doi.org/10.2139/ssrn.1856610