In Cormac McCarthy's post-apocalyptic novel The Road, and the movie adaptation, it's probably safe to assume that the apocalypse was caused by a nuclear war. Considering the story seems to be told only about 10 years after the "cataclysm", wouldn't radiation levels still be dangerously high?
You're thinking of Fallout too much. In the book and the movie a re-occuring description of the damage is gray ash. While this may be like a nuclear explosion, there are other possibilities, such as a solar flare which caused mass fires. I believe in the book (and maybe in the movie) The Man also describes how there were a lot of fires.
I agree completely with Jesse's answer, though I will say that there is a scene (a flashback) where it does seem to imply bombs rather than some natural phenomena. I can't remember exactly where it is, but the man's wife is talking to him, and he hears/sees a thud, and he walks into the bathroom and starts filling the tub with water. I suppose it could be a meteor impact?
The thing that bothered me (tangent here) is that all the higher and lower life forms are dead except people. That's just not realistic. If there isn't enough natural light to support at least some plants, or if the global cataclysm was SO intense that it basically sterilized the earth, then there shouldn't be any people.
I don't believe it was due to a nuclear explosion / war. It makes more sense thinking on what the users on this post mentioned - natural phenomena like a massive volcano is most likely the actual event in the book, and hence in the movie too.
And it makes sense because the man knew exactly what to do when he realized the event has began - he started filling the tub with water right away, instead of running away from there really fast. He knew they were at the right distance to not being hurt, at least not by the event but its effects. Anyway, they had some time to prepare, and even afforded time to get the baby delivered and raised for about 5-6 years (in the movie, the boy is about 10 years, but in the book, the boy is clearly 5 or 6 years old, even if the author does not explicitly said so).