The concept of creating a nuclear firestorm is an old one. Unfortunately, it's also a thoroughly debunked one.
Your attention is directed to this excellent article "(The Impossibility of) Lighting Atmospheric Fire", prepared as coursework by Doowong Chung at Stanford University.
To put it bluntly Chung's article details how this fear of nuclear atmospheric ignition has not only kept on recurring, but how has been constantly been debunked.
Bethe's rebuttal simply refers to and provides an overview of the Los Alamos Laboratory report LA-602 by E. J. Konopinski, C. Marvin, and—oddly enough—Edward Teller (pictured in Fig. 2), the original proponent of the thermonuclear weapon. According to Bethe, although the report was circulated in 1946 and declassified in 1973, "[t]his work was done before the first nuclear test at Alamogordo in July 1945," and its exclusion of atmospheric ignition unaffected by the subsequent development of fusion weapons.  This report, titled "Ignition of the Atmosphere with Nuclear Bombs", gives a detailed accounting of possible energy gain and loss mechanisms that would contribute to—or rule out—a global fusion catastrophe.
This rebuttal was made in 1975 over concerns about thermonuclear weapons. It was not the first and, probably not the last. This was first raised in the 1940s during the Manhattan Project and immediately prior to the Trinity test. Amusingly it was even raised during the development of nuclear weapons by Nazi Germany.
In his memoirs, Albert Speer recounts Heisenberg's evasiveness as to the question of whether fission was guaranteed to be controlled:
Actually, Professor Heisenberg had not given any final answer to my
question whether a successful nuclear fission could be kept under
control with absolute certainty or might continue as a chain reaction.
Hitler was plainly not delighted with the possibility that the earth
under his rule might be transformed into a glowing star. Occasionally,
however, he joked that the scientists in their unworldly urge to lay
bare all the secrets under heaven might some day set the globe on
The report LA-602 looked into the atmospheric ignition in detail. This was in the 1940s.
The report first establishes a few key facts: that detonation of a nuclear bomb "produces a high temperature which will stimulate the reaction of atomic nuclei of the air with each other" and that this will propagate to the entire atmosphere "[i]f an ignition point exists and is surpassed".  This, perhaps trivially, would require "that the energy production in each newly entered region exceed the losses from that region." 
For energy gains, the report chiefly considers runaway
nitrogen-nitrogen reactions, with additional consideration given to
reactions involving protons, as nitrogen nuclei were perhaps the least
stable element present in the atmosphere in significant quantities. In
particular, the reaction that Konopinski et al. saw as "adopting the
most dangerous assumptions" was 
N14 + N14 → Mg24 + α + 17.7 MeV
The energy that results from this reaction is enough to surmount the
Coulomb barriers of the product particles, which is given as
approximately 7 MeV, which, as Bethe explains, means that "the product
nuclei can emerge from the reaction without any difficulty." [8,9]
Due to lack of empirical knowledge of nitrogen-nitrogen cross
sections, the report makes certain simplified assumptions about the
reaction cross section, allowing for an expression for the energy
production rate per nitrogen nucleus, dependent on the temperature.
Konopinski et al. also consider the nitrogen-proton reaction given by
N14 + p → C11 + α + 3.0 MeV
and here too, the produced energy surmounts the Coulomb barriers of
the product particles (given as approximately 2.3 MeV). However, due
to the much lower reaction cross-section and energy yield compared to
the N-N reaction, the report argues that the energy contribution of
this reaction would not be significant.
Essentially the detonation of nuclear weapons is unlikely to ignite the atmosphere to create a nuclear firestorm. If it did, despite the fact this is highly improbable, the nuclear firestorm won't be a slow burn. It will be over very quickly. However, it would require an exceptionally powerful nuclear explosion to ignite the atmosphere. Sufficiently powerful to be considered highly improbable. Perhaps a hypertechnological alien civilization might be able to do it. But that starts getting silly.
This means a nuclear firestorm is off the menu for creating an apocalypse. You may have to consider other possibilities.
For clarification this answer is based on the concept that if a nuclear firestorm cannot be triggered in the atmosphere of planet Earth it cannot happen in the atmosphere of any other life-sustaining planet. Planets with atmospheres dominated by hydrogen such as gas giant planets may be more susceptible to nuclear firestorms, but that will be left as an exercise for the reader. Also, their atmospheres are not life supporting, at least, not in the sense that human life is, and the OP wants human being involved in the apocalypse.