A nuclear detonation only produces significant long-term fallout if its fireball mixes significantly with dirt and debris from the ground. If it does, then the fission products — the split "halves" of the heavy atoms that powered the nuclear reaction — adhere to these relatively heavy particles and "fall out" of the cloud over the course of hours, producing a relatively dense area of "local" fallout downwind.

If a nuclear detonation occurs high-enough that the fireball does not mix with dirt, etc., then the fission products stay hot and light for a longer period of time. These gives the most radioactive (and short half-lived) of them time to burn out, and allows the rest to disperse over a much wider area over the course of days and weeks (and in the case of very large bombs, years). This means that by the time it does "fall out" of the cloud, they are less radioactive in general and more diffused, so no part of the ground gets a large dose of them. This is called "global" fallout (as distinguished from the "local" fallout above) and can be thought of as a slight increase of background radioactivity over an entire hemisphere.

You also get some increased radioactivity at the site of a nuclear detonation through neutron activation, but this is usually relatively short-lived.

There was much to learn about fallout still in 1945 but the scientists were basically aware of the above. The first nuclear test, Trinity, was a tower burst and created, as they expected, a significant plume of radioactivity downwind of it. (Even this was not as radioactive as people came to associate later nuclear fallout possibilities, owing to the fact that the yield of the World War II weapons was a lot lower than the later weapons. We think of fallout in terms of shots like Castle Bravo, which was 1,000X more powerful than the weapon used at Hiroshima, and produced ~700X more radioactive byproducts.)

For the use against Japan, though, they set up their bombs to be high airbursts. This was mainly because they wanted to amplify the blast damage caused by the weapons, through the use of what is called the Mach stem (the initial blast wave is reflected off of the ground, creating a secondary blast wave that interacts constructively with itself, and increases the radius of intense blast damage). But a side-effect of this was that it meant that there would be no long-term contamination in Japan, which was an important thing given that the long-goal of the US was to occupy Japan and turn it into a reliable ally.

The effect that caught them somewhat by surprise, though, was the number of people who died of prompt radiation. Prompt radiation is the initial blast of radioactivity caused by the nuclear detonation (as distinguished from fallout, which is a form of delayed radiation that comes from byproducts). The scientists on the Manhattan Project assumed that if you were close-enough to the detonation to be exposed to a fatal dose of prompt radiation, you'd be close-enough to be killed by the blast and fire effects of the bomb; the latter are generally larger than the former for weapons of this yield (at lower yields, the zone of deadly prompt radiation can be much larger than blast and fire).

But in reality there were several thousand people at each city who somehow survived the blast/fire, but later succumbed to radiation sickness due to prompt exposure. This was unexpected and they initially did not believe Japanese accounts of this, but the Manhattan Project survey team that went in after the Japanese surrendered eventually confirmed this was the case. It should be noted that the highest estimates of how many people might have died purely from radiation are around 20%. Radiation exposure also made the survival rate from other effects, notably thermal burns, worse.

So anyway, to sum up: 1) nuclear fallout was not a goal of the Manhattan Project, 2) nor were radiation casualties in general, 3) they essentially deliberately avoided creating nuclear fallout, 4) but they didn't avoid creating radiation casualties and were surprised that they occurred.

Restricteddata gave a great write up for your answer.

But, in addition to their post, here are some obscure facts:

• Given that the scientists didn't know how bad the possible results of the Trinity test would be, there was an entire division dedicated to medical treatment of the scientists and military and evacuation plans from the test site ready to go at a moment's notice. There were also scouts out in the surrounding communities monitoring the radiation levels. They had been given orders to evacuate the civilians if the numbers reached a certain point.

• There were lead lined tanks created for the sole purpose of Trinity Test retrieval of samples after the day of the blast.

• Extensive training was implemented for the 509th composite group (the ones who dropped the bombs). They were taught to get as much distance as possible after dropping the bombs, except for the scientific monitoring planes following behind them.

Although there were stupid events too.

• collection of trinitite (glass formed from the atomic bomb at the Trinity Test) using a coffee mug.
• the use of these trinitite pieces and uranium cubes as paperweights. Some moldings were made from the same machinery that forged the plutonium hemispheres. -Trinite was sold as jewelry pieces in Times Square until customers reported severe burns on their necks, collars, fingers, wrists and ears.
• the storage of uranium in the captain's quarters upon the USS Indianapolis. -One of the plutonium hemispheres was dropped while being unloaded at Tinian. A military man caught it with his bare hands and suffered from severe radiation burns. These scars stuck with him for the next 75+ years. This gentleman died a year ago and was lucky to have never gotten cancer.

And so much more.