It was possible, but both bombs - Fat Man and Little Boy were in the kiloton range of size, and not like the modern nuclear arsenal that are in megatons of size, so the nuclear fallout at the time was much more limited.

Also, given the prevailing winds flow from west to east, much of the radioactive material fell into the Pacific just east of Japan. Now the rampant nuclear testing (once the scientific community worked out that a nuclear detnation would NOT set the atmosphere on fire - like some suspected it might..) which happened in the Pacific and in the Southwestern US during the 50's and 60's, was an entirely different animal.

That caused a lot of problems in the US and on the Pacific Islands and atolls downwind.

They did not understand radiation sickness and fallout well.

In fact, after the war, the US set up clinics with the ulterior motive of studying the survivors. They talk about it in this documentary, that I’d highly recommend:

https://www.pbs.org/articles/hibakusha-stories-of-survivors-of-hiroshima-and-nagasaki

The two dropped bombs were detonated at altitude. This causes much less radiation near the blast than detonating on the ground because the radioactive contamination is dispersed in the atmosphere rather than in the soil and groundwater.

This is why the destroyed cities didn't have to be abandoned like Pripiat.

When they first conceptualized the bomb some scientists weren't even sure the explosion would stop at all, or if it might create an unstoppable chain reaction that would just continue infinitely and consume the whole earth.

Basically not a real issue. As both detonations were air bursts under 100kt anything bad was swept into the upper atmosphere and dispersed mostly over the Pacific. It did rain back down over years but in miniscule amounts world wide. In short any of those hot particles interacting with life is not great but the concentration and exposure period is what is actually bad. In short there was no fallout risk.

Well, the 300 000 civilians who died because of the bombs weren't involved, so I'm guessing they didn't care.

More of a history question than a science question.

I mean, Western nuclear powers tested pretty close to/deliberately on their own troops and civilians in the coming decades. It's possible the decisionmakers didn't fully appreciate the effect of radiation on health, although the scientists would have certianly known it's dangerous. At best they could have claimed ignorance of the exact long-term effects, and I'm not even sure about that.

Also consider the main landmasses nearby were Korea and China, which the US was also racist against, and which were at the time under Japanese occupation.

Of course the military wasn't worried much. After all, they had just finished the test at Trinity, near Alamogordo, New Mexico, a month before Hiroshima. (July in New Mexico, August in Japan, 1945.)

If they weren't worried in July in the U.S., of course they wouldn't worry about August in Japan. And what neighboring countries were they going to possibly theoretically accidentally hurt? China, Korea, Russia, I guess possibly some Pacific islands ... all of which were considered far less important than the U.S. itself.

Local fallout is dust and ash from a bomb crater, contaminated with radioactive fission products. It falls to earth downwind of the crater and can produce, with radiation alone, a lethal area much larger than that from blast and fire. With an air burst, the fission products rise into the stratosphere, where they dissipate and become part of the global environment. Because Little Boy was an air burst 580 meters (1,900 ft) above the ground, there was no bomb crater and no local radioactive fallout.

https://en.wikipedia.org/wiki/Little_Boy

Fat boy was dropped from 500 meters, so I guess there too was no fallout.

I don't know though if this was known before the drop though.

After the original tests in New Mexico, that risk was seen as very low

Modern nuclear bombs dont have the fallout problems that the ones dropped on Japan had. The WWIi bombs were way smaller than modern bombs. No nuclear bomb in existence can produce anything close to the fallout from Chernobyl. Its got to do with the inefficiency of the reaction.

The US knew there would be fallout ... but had a war to win.