We study an edge demand response problem where, based on historical edge workload demands, an edge provider needs to dispatch moving computing units, e.g. truck-carried modular data centers, in response to emerging hotspots within service area. The goal of edge provider is to maximize the expected revenue brought by serving congested users with satisfactory performance, while minimizing the costs of moving units and the potential service-level agreement violation penalty for interrupted services. The challenge is to make robust predictions for future demands, as well as optimized moving unit dispatching decisions. We propose a learning-based, uncertain-aware moving unit scheduling framework, URANUS, to address this problem. Our framework novelly combines Bayesian deep learning and distributionally robust approximation to make predictions that are robust to data, model and distributional uncertainties in deep learning-based prediction models. Based on the robust prediction outputs, we further propose an efficient planning algorithm to optimize moving unit scheduling in an online manner. Simulation experiments show that URANUS can significantly improve robustness in decision making, and achieve superior performance compared to state-of-the-art reinforcement learning, uncertainty-agnostic learning-based methods, and other baselines.