With all of the secrecy surrounding the Manhattan Project throughout the years-long development process, the atomic bombings of Hiroshima and Nagasaki must have been a complete shock to the world. This sudden demonstration of global warfare supremacy - and the US' willingness to use these weapons against its enemies - is arguably the most significant event in world history.
But when did the American public, and the global public for that matter, truly understand the gravity of the situation, and what one nation's possession of nuclear weapons meant for the future of the world?
You can watch Truman's announcement of the Hiroshima bombing now which, AFAIK, is the first time the existence of nuclear weapons was made public. Now, I'm not sure if the impact of what he’s saying is lost on me or what have you, but I just don’t know if I’d fully appreciate the significance of it if I’d been Joe Citizen in August 1945. Truman equates the power of the bomb to 20,000 tons of TNT, he waxes poetic about harnessing “the basic power of the universe” in a $2 billion gamble, and provides one ambiguous accomplishment in that it “destroyed (Hiroshima’s) usefulness to the enemy.” In hindsight, there’s no question about how momentous this announcement was. But I’m really curious about the reaction it got in the press and your average American’s understanding of what their president had just told them.
At some point in time it became clear that mankind had acquired the capability to destroy itself, and eventually the only thing keeping that from happening was the MAD doctrine. Was this immediately understood by the general public after the bombing of Japan? Was it a creeping realization that settled in over time as further advancements led to more powerful weapons? Did it only really start to sink in following the Soviets' acquisition of nuclear weapons?
The concept of atomic bombs and atomic energy were popularly understood, even scientifically discussed, prior to the attack on Hiroshima. This is not to say that there wasn't shock about it being realized, but that there was significant cultural "priming" is significant for your question. The idea of an atomic bomb, harvesting the basic power of the universe, etc. fit into an existing cultural trope, a trope that included the possibility of civilizational destruction.
Even if someone had not picked that up, however, there was a ravenous news corps willing to fill them in, and a large group of experts who took it upon themselves to evangelize about the new world that humanity was entering into. Core to the interests of many groups, including many of the scientists who worked on the bomb and several of the statesmen who guided it to completion, was the idea that it was a revolutionary break that required radical changes. For the scientists, a world that did not comprehend the gravity of the situation was a world heading towards arms race and eventual destruction. For the statesmen, to contemplate using the atomic bomb as a means of affecting the international order required people agreeing it was not just business as usual.
Over the next few months after the end of World War II, these groups used a variety of means to get their message across, including Congressional hearings, speeches, and even (in the case of the Scientists' Movement) creating new periodicals (such as the Bulletin of the Atomic Scientists) and even hiring lobbyists and advertising experts to help bring their message to the people.
The result was that by 1946, polls of Americans show a remarkable understanding of the "significance." For example, when asked in September 1945 whether they thought that the US could alone monopolize the secret of the bomb, 82% of Americans agreed that it could not be done, with a mere 13% thinking it could be (the remaining 5% had no opinion). 83% also ranked that there was a "real danger" that in the event of another world war, most city-dwelling people on Earth could be killed by atomic bombs. In March 1946, 75% said that should the US fight another war in the next 25 years, it was likely that atomic bombs would be used against it.
This is not to say that these opinions were uniform or without contradictions. The US public was largely proud of having developed and used the bombs. 53.5% of those polled in December 1945 said that had it been up to them, they'd have used the bomb on Japanese cities, just as was done. 22.7% of the same group said they'd have tried to use even more bombs, before Japan had a chance to surrender. One can read these numbers in many ways.
It is important to note that "the significance" of the bomb changed over time, both as there became huge quantities of them available, more than one country who possessed the weapons, and as the means of making more powerful warheads (e.g. the H-bomb) and more sophisticated delivery vehicles (ICBMs, etc.) became available. The atomic bombs of the WWII vintage did not hold out much power of destroying the world — but their successors did. One can ask whether people have even today truly realized the "significance" of these weapons; the experts still debate them.
(MAD comes much later, in the 1960s. You need at least two countries to MAD. It is also a very subtle shift in understanding mutual vulnerability as an asset and not a detriment. Even among high-levels of US military planning, deterrence — much less MAD — was not immediately conceived of. Even today there are those that reject MAD as an approach.)
For poll data, see Hazel Gaudet Erskine, "The Polls: Atomic Weapons and Nuclear Energy," Public Opinion Quarterly 27, no. 2 (Summer 1963), 155-190. For attitudes towards nuclear energy and weapons more broadly, including the pre-Hiroshima "priming" I mention, see Spencer Weart, Nuclear Fear: A History of Images. On changes in doctrine and conceptualizing the bomb from a military and strategic perspective, see Lawrence Freedman, The Evolution of Nuclear Strategy.