GPS receivers have a pretty good TCXO. Usually they are +/- 0.5 ppm spec which are better then ones in most cell phones.
The TCXO does not run continuously and when the GPS receiver is off, the time keeping is a standard 32.768 KHz crystal just like in a PC or digital clock. These are about 100 ppm accuracy.
GPS has many levels time keeping that can accelerate lock times. The first level is being able to predict what satellites are overhead therefore being the best bets to run the correlators against looking for their PRN code. If location is changed significant, like flying on a jet plane to another continent, then this predictiion can actually degrade lock time because many of the satellites may not be in view causing the correlator resources to be wasted.
Part of the satellite location prediction involves setting expected frequency Doppler shift. There can be as much as +/- 3 KHz Doppler shift for satellite coming or going over the horizon. For 1575 MHz, that is equivalent to +/- 1.9 ppm offset in frequency. The digital correlators have a limited frequency lock range of between 300 to 500 Hz so the receiver must 'tune' frequency bins by moving the synthesizer across an assumed frequency error range based on estimating satellite location. If not finding lock after doing all the frequency bins, the software may assume the satellite is not there and may move on, allocating the correlators to another satellite. The worse the time accuracy, the wider the frequency search range must be because of the inaccuracy in predicting the satellite location and its frequency offset.
A GPS receiver keeps track of the last time it was active and computes how wide the time inaccuracy may be based on the 32 KHz clock error. It will widen out its frequency search range the longer it has be turned off to account for the possible time error. This can lengthen time to lock.
For a CDMA cellular network with full GPS assist, satellites visible, precise time to <1 msec, and frequency AFC is used to achieve almost instantaneous lock on satellites. If absolute time is known to better then 1 msec along with satellites health then the GPS receiver does not have to demodulate the 30 baud data which can take several minutes to derive all necessary information to lock on to other satellites. The correlator bandwidth can also be reduced to improve sensitivity.
Newer GPS receivers have more channels and correlators so they can search for more satellites simultaneously and spread more correlator 'traps' over time and frequency to speed the lock process. Tightening down the correlator bandwidth improve sensitivity but increases the number of frequency search bins. Having more correlators offsets the downside of larger quantity of freq search bins, speeding the searching process while accomplishing improved sensitivity.