Accurate time synchronization of the nodes of digital underwater acoustic (UWA) networks is necessary for the effective use of information obtained from various kinds of underwater sensors over a vast water area. The sensors of autonomous underwater vehicles (AUV) are widely used to address new challenges. One of them is time synchronization of sensors on mobile carriers, as well as synchronization of the clocks on mobile nodes of UWA networks, for example, teams of AUVs that perform coordinated and/or cooperated operations. The paper presents theresults of the experiments on using UWA modems for AUV positioning, namely, UWA modems with chip-scale atomic clocks (CSAC) that allow accurate measurements of delays in propagation of UWA signals and, thus, accurate estimation of ranges to their georeferenced sources. In addition, the accuracy of CSAC operation in various situations as well as positioning accuracy of AUVs having UWA modems with integrated CSACs are analyzed based on the experimental results. Also given are practical recommendations on how to "discipline" (steer) CSACs and provide their phase synchronization with a source of timekeeping signals.