There were a variety of responses, as the world is a large and varied place, but a good deal of effort was put into supporting the idea of "international control" — e.g., preventing a nuclear weapons arms race, because it was felt that a world full of multiple nuclear powers was likely to be a very dangerous one for all.

This effort took various forms, though the two official plans discussed at the United Nations. The American approach, the Baruch Plan, was a variation of the Acheson-Lilienthal Report, developed with heavy influence from J. Robert Oppenheimer, the "father" of the atomic bomb. The Baruch Plan basically said that nobody would be allowed to have atomic bombs, and that the UN would be responsible for verifying that this was the case (with lots of inspections) and punishing those who violated it.

It didn't get very far, because it required the Soviets to sign up for all of these inspections before the US would get rid of even one bomb (or stop producing them), and the Soviet didn't want to not have atomic bombs anyway (they were deep into their own secret bomb project at this time).

The Soviet plan, the Gromyko plan, basically just said, "atomic bombs are banned" and didn't really try to enforce or verify that this was actually true. The US appropriately did not trust the Soviets to actually abide by a consequences-free, verification-free treaty.

So international control was a big failure, unless you see the Nuclear Non-Proliferation Treaty as its eventual heir.

On the other hand, for those first few years after the end of World War II, there was a remarkable willingness, even among parties who would later never consider such a thing, to believe that the entire world order was going to have to be remade in order to get more security in an uncertain age. There were those who really believed that the most dangerous world possible was one of secret arms races, and that huge amounts of national sovereignty would need to be surrendered to avoid that. In the end, we ended up with their feared most dangerous world possible, because the amount mutual trust required was too high.

Well this is actually before the nuclear attacks and Japanese surrender (specifically July 24, 1945), but by all accounts of those present, when Stalin was told by Truman of the US's new strategic weapon, Stalin had such a minimal reaction to the news that American and British leaders thought he might not have understood what Truman was telling him at the time (Truman did not refer to it as an atomic bomb, only a very powerful new bomb). Zhukov, who was also present, mentioned in his memoirs, that Stalin's discussion with Molotov immediately after Truman told him indicated that they both knew exactly what Truman was talking about - an atomic bomb.

The relevant excerpts can be found here.