In the early days of the Manhattan project, there was a brief scare, based on some calculations by Edward Teller, that an atomic bomb would ignite the atmosphere. They later redid the calculations and concluded that the probability was very small (something like one in three million).
What if we lived in the universe where that improbable event happened, and the Trinity test ignited the atmosphere in the way that Teller predicted?
- Could somebody in America get word out to the rest of the world, just before being incinerated?
- Would the wavefront of burning atmosphere move slowly enough that people in, say, India had enough time to pull together a few people, and put them somewhere where they might have a shot at survival (at least for more than a few days)? And what place could they use?
- Is there a reasonable scenario where they could survive for say a few months or a year, and use that time to figure out a way survive in the now absolutely barren Earth (would the atmosphere be gone, or just changed into something entirely different).
- If they could manage cling on to their miserable existence and generate offspring, is there some way they could restore the atmosphere, over many generations?
- If not, and humanity is doomed, would life survive on earth?
Since the poor souls in this universe were very unlucky, we can allow them some luck in the little time they have to save themselves: being in the right time at the right place, and so on. The odds are obviously against humanity, so they might catch a little break here and there, but we assume only one spectacular coincidence.
EDIT: I'm going to slightly move the goalposts here (with apologies to the authors of the existing answers), because we're getting into discussions about the nuclear physics, which I had hoped to avoid.
So let's add the following: the test causes a very, very slightly self sustaining nuclear reaction. Technically, this would increase exponentially, but the rate is so slow that the result looks more like the world being on fire than the world blowing up. Can some quick-thinking American get word to some quick thinking Indian, ahead of the wavefront of the fire?
Is there somewhere people with advanced warning can seal themselves off with a supply of oxygen and some plants? There's no need to make the point that survival is very unlikely, but given a little luck, is there a way that we could just pull through, or at least cling on for a few more weeks.
EDIT2: To be even more blunt: this question is not about physics, it is about logistics. The Trinity test just gives us a convenient point in time (ie. level of technology), and a basic mechanism to work off to set the parameters. The question is, if the atmosphere were on fire, would there be some way of surviving. How would we get the message out of America? How much time would people in Europe and Asia have? What structures existing in 1945 would allow one to survive until the atmosphere had burnt up. If you're in India, and you get word that the world will be in flames in an hour, what actions should you take?