Online action detection aims at identifying the ongoing action in a streaming video without seeing the future. Timely and reliable response is critical for real-world applications. In this paper, we introduce Bayesian Evidential Deep Learning (BEDL), an efficient and generalizable framework for online action detection and uncertainty quantification. Specifically, we combine Bayesian neural networks and evidential deep learning by a teacher-student architecture. The teacher model is built in a Bayesian manner and transfers its mutual information and distribution to the student model through evidential deep learning. In this way, the student model can make accurate online inference while efficiently quantifying the uncertainty. Compared to existing evidential deep learning methods, BEDL estimates uncertainty more accurately by leveraging the Bayesian teacher model. In addition, we designed an attention module for BEDL that can select important features based on the Bayesian mutual information for online inference. We evaluated BEDL on benchmark datasets including THUMPS'14, TVSeries, and HDD. BEDL achieves competitive performance while keeping efficient inference. Extensive ablation studies demonstrate the effectiveness of each component. And the uncertainty quantification is verified by experiments of online anomaly detection using the student model.