The question has the facts slightly confused. It was Edward Teller, always an enthusiast for very large explosions, who raised the question. This wasn't a matter of ordinary combustion, because trying to burn nitrogen with oxygen doesn't release heat, but absorbs it. That's easy to determine: the atmosphere has not gone up in flames over millions of years of lightning strikes.
The actual fear was of a nuclear chain reaction, fusing the nuclei of nitrogen atoms. If this could be started and become self sustaining, then the entire atmosphere would go up in a nuclear explosion. This would certainly wipe out all life on Earth. However, this is physically impossible.
The mathematics and physics to calculate this was available at the time and Hans Bethe and Emil Konopinski did the calculations and showed that it was impossible, with a large margin of error. A few years later, Konopinski, C. Marvin and Teller did the calculation for thermonuclear weapons, and showed that it was impossible for them, too. In fact, their report is available on-line here.
To summarise, the heat produced by such fusion is radiated away much too fast for the reaction to be self-sustaining. You probably will get some nitrogen fusion within a nuclear explosion, but the reaction stops very quickly. Practical fusion explosions need their fuel to be at vastly greater density than the atmosphere, and to be confined in some way. These conditions just aren't available for a bomb set off in the atmosphere.
So, this was not a concern at the time of the first nuclear test (which was not dropped, but set off on top of a tower). Enrico Fermi was offering to take bets on its likelihood, but this was a joke.