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Multi-access edge computing (MEC) is viewed as an integral part of future wireless networks to support new applications with stringent service reliability and latency requirements. However, guaranteeing ultra-reliable and low-latency MEC (URLL MEC) is very challenging due to uncertainties of wireless links, limited communications and computing resources, as well as dynamic network traffic. Enabling URLL MEC mandates taking into account the statistics of the end-to-end (E2E) latency and reliability across the wireless and edge computing systems. In this paper, a novel framework is proposed to optimize the reliability of MEC networks by considering the distribution of E2E service delay, encompassing over-the-air transmission and edge computing latency. The proposed framework builds on correlated variational autoencoders (VAEs) to estimate the full distribution of the E2E service delay. Using this result, a new optimization problem based on risk theory is formulated to maximize the network reliability by minimizing the Conditional Value at Risk (CVaR) as a risk measure of the E2E service delay. To solve this problem, a new algorithm is developed to efficiently allocate users' processing tasks to edge computing servers across the MEC network, while considering the statistics of the E2E service delay learned by VAEs. The simulation results show that the proposed scheme outperforms several baselines that do not account for the risk analyses or statistics of the E2E service delay.