Mobile Underwater Internet of Things (UIoT) has become one of the pivotal technologies in the advancement of smart oceans monitoring systems, with autonomous underwater vehicles (AUVs) offering a highly promising way for collecting data from mobile sensor nodes. In Mobile UIoT, the mobility of sensor nodes leads to dynamic changes in network topology, making it difficult for the network to form stable clusters. The selection and position of cluster-head nodes will also dynamically change, which increases the difficulty of AUV collecting cluster-head nodes data and further increases data collection delay. To overcome these above difficulties, AUVs are employed as mobile edge devices for collecting data from nodes within the UIoT. In addition, a data collection algorithm orientated for dynamic network topology (DCADNT) is proposed in this paper, which includes three phases: dynamic clustering, dynamic transmission, and dynamic data collection. DCADNT not only considers the impact of dynamic topology on data collection, but also takes into account the impact of propagation delay. By calculating the packet delivery probability based on propagation delay, the reliability of data transmission can be effectively guaranteed. The performance of DCADNT is evaluated by extensive simulations and the compared results with other typical data collection algorithms are given, which demonstrate that DCADNT effectively reduces data collection delay and improves packet collection rate.