story identification - Fantasy book that turns out to be post-apocalypse, set several millennia after a nuclear war

This sounds like it may very well be The Sword of Shannara series by Terry Brooks.

I remember reading it and feeling it was a basic fantasy, until at one point the story moves to the ruins of an ancient city. If I recall correctly, a monster that was an ancient robotic device then menaces them, at which point I understood that it really was a post-apocalyptic scenario.

It's been many, many years since I read it, but the wikipedia entry does mention the background of the races and the general post-nuclear loss of technology, although it does explicitly mention that magic is a part of the story:

The Sword of Shannara's events take place 2000 years after an apocalypse has occurred: nuclear holocaust has wiped out most of the planet. During this time, Mankind mutated into several distinct races: Men, Dwarves, Gnomes, and Trolls, all named after creatures from "age-old" myths. Also, the Elves begin to emerge after having been in seclusion and hiding for centuries. The warring that caused the holocaust is referred to as the "Great Wars" throughout the novel. These wars rearranged the planet's geographical attributes and wiped out most life forms on Earth. As a result of the Great Wars, most advanced technology has been lost, thus most of the events in the Shannara series take place in a medieval setting. However, magic is also back into the world, rediscovered after the loss of technology.