The setting of "Coming Attraction" can be best described as post-apocalyptic, although "apocalypse" might be too strong of a term to actually apply to the relatively intact and functional society that New York has become after the Hellbomb. There are still all the normal trappings of prewar society, as we understand them in the real world, but the lingering effects of the bomb hover visibly and figuratively over the town and cast a poisonous pall over the story.
This setting, however, is not crucial to the plot, and yet is crucial to the story. The plot, at its core, could happen anywhere, at any time, but its exact execution and deeper meaning rely upon the setting.
Consider, for example, the effect of the radiation in the story. Radiation is used both literally and symbolically; Turner wears a film jacket that he uses to determine his level of radiation exposure, and he sees and emotionally responds to the signs of radiation around the city, but radiation also implies a pervasive, invisible taint. New York is both literally and figuratively poisoned, and the characters, some without realizing it, respond as such.
In my interpretation, the effect of the setting is to accentuate and exaggerate real-world behaviors, basically to make not-so-subtle elements of American culture even more obvious, to the point of perversion. For example, the woman's use of a mask, coupled with a "Cretan Revival" breast-baring bodice; this suggests an evolution of the real-world American attitude toward sex (consider the prevalence of sex in our marketing but the scandalous way it is treated when actually performed).
The extension of the war to include the moon is a good example of this as well; it seems ridiculous that the war would need to extend into space, or that the American woman would say that the moon is "ours and Russia's", but this suggests the lengths to which the American ego has gone to puff up its importance, ignorant of the destruction it has caused.
The American characters, in general, seem to go about their lives as they would have otherwise; it is only their behaviors out of this context that strike us as wrong, and this is due to the setting. The plot itself, though, is not dependent upon the setting. Turner's reason for being in America, trade, is mundane. The woman's motivation for tricking him, emotional fulfillment, is mundane as well. Neither person has any dependence or interaction with the setting that fundamentally drives the story.