I don't know what chain reaction exactly they were thinking of, but from modern fusion research, I believe we can confidently say that the atmosphere would need to be interior-of-a-large-star-level dense, and even then I'm not sure you'd get nitrogen fusing with anything without a lot of hydrogen or helium around. Nitrogen-nitrogen fusion seems extremely implausible for sure
Prolly the most relevant paragraph from the linked article for this discussion:
Today, especially after the detonation of the 50 MT Tsar hydrogen bomb on Novaya Zemlya in 1961, it is also experimentally verified that the danger of atmospheric or even oceanic ignition does not exist. Also, the experimental measurements obtained by Zucker and others demonstrate that the fusion probability is much smaller than the geometric cross-section for 14N+14N assumed by Teller and coworkers, further reducing the chances for such an event. Furthermore, the atmosphere is also heated only to temperatures of a few million degrees, so that the most efficient energies of the fusing nuclei are a few 100 keV and thus well below the Coulomb barrier and very much reduced by penetrability. These temperatures are noticeably lower than those in the late hydrostatic burning stages of massive stars.
Basically the temperature of the atmosphere is over an order of magnitude too low to have any chance of ignition (need 10s of millions of K), and the reaction rate is thus several orders of magnitude lower than the threshold.
i think the idea is that the part that already fused creates a blast wave that could create the conditions, including preassure required for more fusion. i have no idea if it's possible though.