I know the principle of nuclear fission had been widely understood for a decade or more by the scientific community, but the idea of a nuclear bomb was presumably a much more theoretical concept in the minds of most nuclear scientists.
What was the initial Japanese reaction to the bombs from a scientific perspective? Did they more or less immediately realise they were atomic? Did the US give any indication? Were Japan aware of the long-term consequences of nuclear fallout once the initial explosions had ended?
The Japanese high command learned about bombing of Hiroshima from US propaganda announcements about it; the city had been cut off from communications because of the bombing so they got no direct news. Their first order of business was to send a team of scientists to confirm whether the claim was real or not; some believed it was a lie, some suspected it was likely true. (They knew about nuclear fission, but it was still hard to believe anyone had pulled it off. To be sure, the German scientists who were told about Hiroshima at Farm Hall also initially disbelieved it.)
Dr. Yoshio Nishina, who had headed part of the Japanese fission research project, was sent to Hiroshima. He arrived on the morning of August 8th. By examining the wreckage of the town, the human remains, and taking radioactivity readings, he concluded that the US was telling the truth: they had detonated a single, extremely-large bomb that released a lot of radioactivity. He sent the following account back to the high command on the evening of August 8th:
What I've seen so far is unspeakable. Tens of thousands dead. Bodies piled up everywhere. Sick, wounded, naked people wandering around in a daze... Almost no buildings left standing. It's all true then? Hiroshima is completely wiped out? Completely. ... I'm very sorry to tell you this... the so-called new-type bomb is actually an atomic bomb.
There is a lot that happened afterwards (see this previous post of mine which goes into the "timeline" in detail), but after that point the Japanese high command doesn't seem to have ever doubted that it was a real atomic bomb.
The fallout question is more complicated. There wasn't really any significant fallout from the bombs, because they were detonated at such a high altitude. There was a lot of exposure of people to radioactivity, however. Acute radiation syndrome was observed by doctors in Hiroshima and Nagasaki, and reported as the "atomic plague." This was at first vigorously denied by the Americans, who feared that the Japanese were using these kinds of claims as propaganda (e.g., that they would be seen as victims), and didn't believe it was likely that someone would get exposed to a fatal dose of radiation but survive the other effects (that is, they thought the blast/fire would kill anyone who got that much radiation). But after V-J day, the US sent scientists as part of the first landings of the Occupation soldiers and they confirmed, over time, that this had been true, and that some 15% or so of the fatalities were due to radiation alone.
After the war, the US and Japanese jointly kept records of the people in the cities as the Atomic Bomb Casualty Commission, and used the exposures in part as a way of understanding in more detail the long- and short-term effects of this kind of radiation exposure.
The Nishina quote comes from Monica Braw, The Atomic Bomb Suppressed.