The hazards of radiation were discovered well before atomic weapons were. Radiation was discovered in the late 19th century and by the early 20th century had been widely popularized, written about, talked about. The dangers of excessive use of unshielded X-ray machines were discovered soon afterwards, as were negative health effects from consuming radioactive materials. The case of the Radium Girls brought the issue lots of popular and scientific attention in the 1920s.

If you mean, specifically, that atomic bombs could cause radiation poisoning — the issue here is not that people did not think they could do that, but that the early scientists did not think they would do that. Not because they were safe, but because they were so dangerous in other ways. They judged it likely that anyone who would get a severe radiation dose would probably already be dead from fire and blast effects. They were wrong; life is more complicated than that, and a significant number of people survived the non-radioactive effects to perish or become ill from the radiation effects.

The initial reports of radiation sickness by Japanese scientists were thought to be just sympathy-generating wartime propaganda. In the fall of 1945, however, the US sent scientists to investigate the effects of the bombs, including medical effects, and concluded that there were indeed radiation-induced effects among the victims. They admitted such in public reports and hearings, although they often downplayed them. (They were not entirely incorrect to downplay them — they are not the cause of the majority of casualties, even though they are exotic and thus more interesting to use than burns or blast injuries.)

As for change of policies, there was no real policy that existed at this point, so it didn't have any chance to change them. The more important radiation-related policy issue was the discovery that very high yield nuclear weapons (i.e. thermonuclear weapons) produce very large zones of radioactive fallout if detonated on the ground. This was discovered only around 1954 or so. The major policy change that resulted was not so much in the use of the weapons, though it had implications for that (you don't want to blanket an ally, or your own troops, with fallout), but in the defense against them — specifically, it ushered in an era of emphasis on fallout shelters (places to wait out radiation) as opposed to blast shelters (places to avoid having things fall on you).

On public reactions, generally the public (of all countries) has been uniquely focused on radioactivity as a special, unusual, exotic hazard. It is what risk analysis people call a "dread risk" — it is something mysterious and unknown and dips its hand into deep psychological waters (e.g. it messes up genes which can produce birth defects which makes people associate it with creating "monsters"). It arguably does not deserve so much focus, because it is just one risk in an array of risks and often not as big as risk as many other risks. (Risks that are more "familiar" are often discounted by people — this is why many people are afraid to live next to nuclear plants but are not afraid of driving their cars at night, even though the latter poses a vastly larger risk to their health.) The history of this is hard to sum up in a paragraph, but the basic gist of it is that the imagery of radiation has been a tricky thing for both scientific commentators and governments to get a handle on. At times they have tried to mobilize it, or combat it, to various ends, but it is not something they are entirely in control of, and often leads to public opinions that they find inconvenient. This is one of the reasons that scientists and governments have often been accused (sometimes correctly) of covering up or whitewashing radioactivity concerns, because they fear that in the hands of the "uneducated" it poses too great a threat. And of course, when they do cover it up or downplay it, that only feeds the idea that there is something sinister to be feared.

On the initial reaction to radiation sickness reports, see here. On early reactions to radiation hazards, pre-bomb, as well as later fallout concerns, and the cultural ramifications of radioactivity, Spencer Weart's The Rise of Nuclear Fear is a standard reference, and a great read. On "dread risks," see the work of Paul Slovic.