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 corre

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.