To improve channel estimation (CE) for reconfigurable intelligent surface (RIS)-assisted systems in time-varying mmWave channels, this paper proposes a two-stage adaptive CE scheme. This is the first attempt to develop a CE scheme without assumptions on specific timescales for channel variations. In the first stage, the adaptive scheme incorporates the estimation of partial channel state information and a channel status check process. The introduced check process can monitor the changing status of the channels and provide information for the second stage. In the second stage, based on the results from the check process, the adaptive scheme adaptively selects from two proposed candidate CE algorithms: Two-Phase orthogonal matching pursuit (TP-OMP) and Structured-Shift OMP (SS-OMP). Simulation results show that both TP-OMP and SS-OMP can reduce pilot overhead by around 33%, and respectively lower the computational complexity of existing works by about 55% and 65%. Additionally, the check process obtains an accuracy rate of approximately 92% so that the proposed CE scheme can maintain stable CE performance in time-varying channels.