There were definitely dangers of meltdowns — the reactors being developed were all pretty much first-attempts (with barely any prototyping) and were not built with many of the safety features which would be considered standard today (like containment domes). The level of uncertainty in their operation was high — they were truly venturing into new territory, and at times that was made especially clear. The three industrial-sized reactors at Hanford site in Washington State were built at enough of a distance from one another that if one was damaged by its operation they could imagine keeping the others functioning. Problems with the neutron irradiation of the carbon moderator forced the US to run the reactors at relatively low power in the postwar period, because to do otherwise was to create serious safety risks (the moderator was swelling, and that could complicate the action of both the fuel and control rods; they eventually figured out a technical solution to the problem but it took some time). The people working on the reactors thought there were many risks involved.

Separately, there were also risks of accidental criticality excursions at all of the sites that handled fissile material (which included Hanford, Oak Ridge, and Los Alamos). There were safety groups that attempted to minimize these risks but there were again many unknowns. The properties of a critical system are dependent on a lot of variables, including (but not limited to) geometry, the presence of potential moderators (like water), and enrichment levels. At Oak Ridge it was especially tricky because uranium of different levels of enrichment had to be circulated and stored under quite a lot of conditions, creating a lot of different possibilities for danger.

There were also deliberate experiments (criticality experiments) that purposefully came very close to setting off uncontrolled reactions, in the name of figuring out the parameters of the critical mass. There were several accidents in this field, two of them being fatal (Harry Daghlian and Louis Slotin).

There were real risks of accidental detonation at the Trinity test, as the electrical system for the assembled "Gadget" could be induced by lightning strikes and other accidental charges. This was not an ideal fear; a model of the electrical system actually did unintentionally fire (fortunately not hooked up to a bomb) during a storm.

There were also fears that the planes taking off with the bombs, on the island of Tinian, could crash and trigger an atomic explosion or at least a dispersal of fissile material.

I think the most debilitating possible event would have been something bad happening with the plane on the island of Tinian, because the Tinian base itself was extremely crucial to the effort against Japan, and hampering its efforts or contaminating it (much less nuking it) would have really gummed up a lot of things. Reactors melting down would have seriously hampered the effort to develop plutonium, but otherwise would not have likely affected the overall war effort. Accidents at Oak Ridge might have hampered the uranium effort, but that's it. Large releases of nuclear material (say, by the meltdown of a Hanford reactor) would make keeping the project secret very difficult (they would be relatively easy to detect and analyze off-site). Criticality accidents at Los Alamos could be fatal and damage their work there, but would be limited in scope, essentially.