We are never told precisely when the novel is set, however, from the descriptions of the devastation that some kind of apocalypse has caused, and from the flashbacks that the novel includes, we can infer that it is set in the near future, as people lived their lives as they do today. Note the description of the house that they enter where the father grew up:

They walked through the diningroom where the firebrick in the hearth was as yellow as the day it was laid beause his mother could not bear to see it blackened. The floor buckled from the rainwater. In the livingroom the bones of  small animal dismembered and placed in a pile. Possibly a cat. A glass tumbler by the door.

We can see that houses in the novel, apart from the obvious destruction, are how we expect houses today, which suggests that the novel is set in some kind of immediate future. Note how this setting increases the shock impact of the novel. Again and again, the contents of the novel, as horrific as they are, are not completely removed into the realm of fantasy. This could happen and we would probably react in the same way, the setting of the novel seems to stress. We too could descend into a complete anarchic state after the end of civilisation thanks to some kind of dystopian disaster.