Mobile edge computing (MEC) is a promising technology for enhancing the computation capacities and features of mobile users by offloading complex computation tasks to the edge servers. However, mobility poses great challenges on delivering reliable MEC service required for latency-critical applications. First, mobility management has to tackle the dynamics of both user's location changes and task arrivals that vary in different timescales. Second, user mobility could induce service migration, leading to reliability loss due to the migration delay. In this paper, we propose a two-timescale mobility management framework by joint control of service migration and transmission power to address the above challenges. Specifically, the service migration operates at a large timescale to support user mobility in the multi-cell network, while the power control is performed at a small timescale for real-time task offloading. Their joint control is formulated as an optimization problem aiming at the long-term mobile energy minimization subject to the reliability requirement of computation offloading. To solve the problem, we propose a Lyapunov-based framework to decompose the problem into different timescales, based on which a low-complexity two-timescale online algorithm is developed by exploiting the problem structure. The proposed online algorithm is shown to be asymptotically optimal via theoretical analysis, and is further developed to accommodate the multiuser management. The simulation results demonstrate that our proposed algorithm can significantly improve the energy and reliability performance.