The particular fear that had been voiced, very early on in the project, was that the nuclear fission reaction that powered the bomb could produce enough heat to generate nuclear fusion reactions in parts of the atmosphere. Specifically they were worried about nitrogen fusing with other nitrogen.

They were able to determine fairly quickly that this was unlikely for several reasons. First, while an atomic bomb is indeed very hot, it is almost nothing compared to certain events in Earth's geological past, which included asteroid strikes and the like. If that much heat could cause a run-away fusion reaction, then we wouldn't be here already. So that's an indirect way to reason that this won't happen.

They also studied the problem with some interest. They didn't know exactly what conditions were required to make nitrogen fuse with itself — they didn't know the fusion cross-section, in technical terms. What density of nitrogen, heated to what temperature, would allow for nuclear fusion? Nuclear fusion is not easy — atoms don't "want" to fuse. It turns out that even if you really want to use an atomic bomb to make fusion reactions happen, it is very difficult (as they would discover when they tried to make thermonuclear weapons).

Even without knowing the nitrogen-nitrogen reaction conditions, they could guess at them, based on other knowledge they had. So they said, what if we take our best guess, and then imagine that it is several orders of magnitude more likely to happen than we might imagine — they were being conservative about this. They found that if you made that assumption and then did the math, you didn't get any reactions. Or, to put a very specific point on it, even if you did get a few fusion reactions, it wouldn't be enough to propagate: the heat necessary would not be maintained, and the reaction would cool and die out.

Even this was somewhat tentative, because again they didn't know either the exact conditions that were going to be produced by an atomic bomb (they'd never set one off before) nor the exact conditions necessary for fusion (they hadn't done it before). But they reasoned that these conclusions, combined with the fact that it had never happened in the past under even hotter conditions, made them very confident that it wasn't going to happen.

So by the time of the Trinity test this was not a serious scientific concern. But of course once you have heard of such a possibility, it's hard to get it out of your head, and there were people at the test who reported having a lingering fear of it.

For more details about the calculation, and what it would actually require to turn the planet into a thermonuclear cinder (in the 1970s, weapons scientists did that calculation as well, out of curiosity), you can see this blog post I wrote a few years back.