Not right away, it will just be slightly less accurate. Ignoring all other sources of errors, for extremely high precision work you still might want to correct for the relativistic difference in time between the satellite and the surface; but in a post-apocalypse setting, I don't think you'll mind the small drifts.
There's still plenty of other maintenance, like delta-v maneuvers that need to get done or the system will eventually fail. Likely within a year.
In japan, in the typical post-apocalypse setting, small drifts are indeed not a problem. Who cares if your GPS has drifted by a few centimeter? That apocalypse may have moved the country by meters.
This system will probably be useful for quickly determining what moved in an earthquake.
There's still plenty of other maintenance, like delta-v maneuvers that need to get done or the system will eventually fail. Likely within a year.