High-speed mobile underwater acoustic communication (HSM-UAC) is a relatively unexplored terrain, but its importance is expected to increase with the widespread application of flexible underwater mobile platforms such as AUVs and UUVs. Knowledge of HSM-UAC channels is a prerequisite to achieving reliable communication. This work conducted two HSM-UAC channel measurement experiments with a maximum speed of 20knots. The analysis of channel time-varying characteristics demonstrates the shortcomings of traditional consistent Doppler frequency offset channel model. The experimental findings reveal the presence of two non-stationary modes in the HSM-UAC channel. For short moving distances, the changes in channel geometry result in each path experiencing a different Doppler scale, thus observing the multiscale-multilag (MSML) channels. For long moving distances, the alteration in sound ray propagation tracks entirely reshapes the channel impulse response (CIR) structure, leading to a rapid decline in channel time correlation. In further analysis, the correlation coefficient of the power delay profile (PDP) is used to measure channel stationarity. Subsequently, the stationary time and corresponding stationary distance are calculated as characterization parameters for the time-varying channel model. The quantitative evaluation results are helpful for the communication signal design and determining the update frequency of channel state information.