Mobility performance analyses of base station (BS) cooperation have an important role in the design of BS cooperation strategies for mobile UAVs in a cellular-connected UAV network (CCUN) for Internet of Things applications. This paper presents an in-depth study on the mobility performance analyses of a mobile UAV under BS cooperation in a CCUN. Performance models are derived for analyzing the mobility performance of a mobile UAV under BS cooperation in terms of the handover rate, handover probability, coverage probability, and average throughput, taking into account the occurrence of a handover. In deriving the performance models, modified 2-D and 3-D random waypoint (RWP) mobility models are used to describe the random or uncertain flight trajectories of a mobile UAV during a task execution. Based on the derived performance models, the impacts of main network parameters on the mobility performance of a mobile UAV under BS cooperation are investigated, which can help determine reference values for the network parameters in the design of a BS cooperation strategy for a mobile UAV. The difference between the mobility performance of a mobile UAV under a 2-D RWP model and that under a 3-D RWP model is investigated, which reveals several useful results for the design of BS cooperation strategies for mobile UAVs.