Having good knowledge of terrain information is essential for improving the performance of various downstream tasks on complex terrains, especially for the locomotion and navigation of legged robots. We present a novel framework for neural urban terrain reconstruction with uncertainty estimations. It generates dense robot-centric elevation maps online from sparse LiDAR observations. We design a novel pre-processing and point features representation approach that ensures high robustness and computational efficiency when integrating multiple point cloud frames. A generative Bayesian model then recovers the detailed terrain structures while simultaneously providing the pixel-wise reconstruction uncertainty. We evaluate the proposed pipeline through both simulation and real-world experiments. Our approach achieves high-quality terrain reconstruction with real-time performance on a mobile platform, and the uncertainty estimates may further benefit the downstream tasks of legged robots.