I'm not an expert on the physics of the GPS system, but I believe it relies upon accurate data regarding the position of each satellite and the propagation delay of the signals generated by each satellite.
Since the radio signals from the satellites move at roughly one nanosecond per foot (that is, one billionth of a second per foot) and since a millisecond is equivalent to one million nanoseconds, one might have to deal with errors on the order of one million feet if the timekeeping is off by a millisecond.
My GPS makes errors in guiding me, but an error of a million feet would make the system quite useless.
“Since the radio signals from the satellites move at roughly one nanosecond per foot (that is, one billionth of a second per foot) and since a millisecond is equivalent to one million nanoseconds, one might have to deal with errors on the order of one million feet if the timekeeping is off by a millisecond.”
GPS clocks are synched to each other and not to UTC.