We consider the problem of adaptively stabilizing linear plants with unknown time-varying parameters in the presence of noise, disturbances, and unmodeled dynamics using the supervisory control framework, which employs multiple candidate controllers and an estimator based switching logic to select the active controller at every instant of time. Time-varying uncertain linear plants can be stabilized by supervisory control, provided that the plant's parameter varies slowly enough in terms of mixed dwell-time switching and average dwell-time switching, the noise and disturbances are bounded and small enough in terms of L-infinity norms, and the unmodeled dynamics are small enough in the input-to-state stability sense. This work extends previously reported works on supervisory control of linear time-invariant systems with constant unknown parameters to the case of linear time-varying uncertain systems. A numerical example is included, and limitations of the approach are discussed.