What is the setting of "By the Waters of Babylon"?

For the setting of "By the Waters of Babylon," you will have to give generalized details, not specifics, because no specifics are given in the text at all.  We can narrow it down to a pretty precise location, but not time period.

For a time period, all that we know is that it is set in the distant future.  Given that the story gives clues that suggest a nuclear bomb wiped out most of civilization, it had to be after the 1940's, when nukes were first developed and tested.  Then, mankind is practically wiped out in the wake of those bombs, and at least several generations later, the story begins.  John and his father don't remember first-hand what happened in the "great burning," so it is at least 50 years afterwards, and maybe longer, since John's tribes are large and well-established in their traditions.

For the actual location, we can take clues from the city of the Gods.  He passes by a statue that says "Ashing," which we can infer was a statue of Washington.  Also, there are large skyscrapers, subway systems, and large highways and streets.  There is a river too. It is probably in Washington D.C. where many national monuments are kept, like statues of Washington, and where rivers and large cities are housed.  So that is what we can get for the actual location.

I hope that those thoughts helped a bit; good luck!