Within hours of the Hiroshima bombing the US began flooding the world with press releases/propaganda/news articles about the use of the atomic bomb. This was a very carefully planned campaign that had been developed by the same people who made the atomic bombs, with several goals in mind, including convincing the Japanese that the bombs were real, and impressing upon them the "fantastic" nature of the weapon in the hopes that it would convince them to surrender (that is, they did not want them to think of it as "just another weapon").

Included in these statements/stories was information about how the bomb was made, how it basically worked (on a very basic scientific level), and an account of the test of the first one. Again, the goal was to provide the evidence to make this fantastical claim ("atomic bombs") seem real.

The Japanese had access certainly to the broadcast propaganda. Understandably, given the context of wartime, they did not trust it on its own, and ordered that a team of scientists go to Hiroshima to see if the claims could be substantiated with readings on the ground. The results from this (that it was an atomic bomb) were transmitted back on the evening of August 8th. (That night, the Soviet Union declared war on Japan and invaded Manchuria. On the morning of August 9th, the Japanese cabinet met to discuss these matters. During this meeting, they got news of the Nagasaki attack.)

As for the question of stockpile size and production rates, these were considered amongst the highest secrets of the project. They were not broadcast. It is not quite right to say that the US "only made three for the first round" — better is to say, "they only had enough bomb fuel for three weapons by the end of July 1945." Their production system was still going in full-swing, however, and they would have had another bomb ready to use by August 17th or so, and three to four bombs a month after that, maybe even more if the war had gone on and they had modified the bomb designs to be more efficient in their use of fuel (which was on the table). I only bring this up because the idea that the US "bluffed" Japan is not really true; they would have been able to drop an atomic bomb about once a week after August 10th or so.