Was It Actually Possible For A Nuclear Bomb/Explosion To Set The Atmosphere On Fire?

Actually, no. In summary, two of the reasons are that so much of the energy are lost into radiation and light.

It is shown that, whatever the temperature to which a section of the atmosphere may be heated, no self-propagating chain of nuclear reactions is likely to be started. The energy losses to radiation always overcompensate the gains due to the reactions.

It is impossible to reach such temperature unless fission bombs or thermonuclear bombs are used which greatly exceed the bombs now under consideration. But even if bombs of the required volume (i.e., greater than 1,000 cubic meters) are employed, energy transfer from electrons to light quanta by Compton scattering will provide a further safety factor and will make a chain reaction in air impossible.

Later:

If, after calculation, [Compton] said, it were proved that the chances were more than approximately three in one million that the earth would be vaporized by the atomic explosion, he would not proceed with the project. Calculation proved the figures slightly less — and the project continued.

In later studies, it was determined to actually be impossible.

Reference: https://www.insidescience.org/manhattan-project-legacy/atmosphere-on-fire

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