As one of the most popular consumer electronics, Unmanned Aerial Vehicle (UAV) has the potential to assist User Equipment (UE) in the Internet of Everything. Mobile edge computing (MEC) is also an emerging technique that can provide sufficient computing resources to IoE users. In this paper, we focused on the UAV-assisted MEC problem with mobile UEs, where the UAV serves as an MEC server to assist UEs in computing and acts as a relay to deliver tasks to a ground access point (AP). The objective is to minimize the average time for completing all tasks in the network while jointly optimizing resource allocation such as communication bandwidth, CPU frequency, task division ratio, and UAV’s three-dimensional location deployment. First, we proposed an online MEC network adjustment scheme. Then, we decomposed the formulated non-convex problem into three low-complexity subproblems. Last, we proposed a successive convex approximation-based joint optimization algorithm to solve them. In addition, we presented heuristic conclusions in task allocation rules, UAV flight trajectory prediction, and UAV hovering altitude selection, which can further reduce task completion time and deepen the understanding of the working mode of mobile edge servers. Simulation results show that the algorithm can significantly reduce the completion time.