Trajectory prediction is an essential component of the next generation national air transportation system. Reliable trajectory prediction models need to consider uncertainties coming from multiple sources. Environmental factor is one of the most significant reasons affecting trajectory prediction models and is the focus of this study. This paper propose an advanced Bayesian Deep Learning method for aircraft trajectory prediction considering weather impacts. A brief review of both deterministic and probabilistic trajectory prediction methods is given, with a specific focus on learning-based methods. Next, a deterministic trajectory prediction model with classical deep learning methods is proposed to handle both spatial and temporal information using a nested convolution neural network, recurrent neural network, and fully-connected neural network. Following this, the deterministic neural network model is extended to be a Bayesian deep learning model to consider uncertainties where the posterior distributions of parameters are estimated with variational inference for enhanced efficiency. Both mean prediction and confidence intervals are obtained giving the last on-file flight plans and weather data in the region. The proposed methodology is validated using air traffic and weather data from the Sherlock data warehouse. Data pre-processing procedures for big data analytics are discussed in detail. Demonstration and metrics-based validation are performed during severe convective weather conditions for several air traffic control centers. The results show a significant reduction in prediction variance. A comparison with existing methods is also performed. Several conclusions and future works are given based on the proposed study.