Offline reinforcement learning (RL) provides a framework for learning decision making from offline data and therefore constitutes a promising approach for real world applications as automated driving. Self-driving vehicles (SDV) learn a policy, which potentially even outperforms the behavior in the sub-optimal data set. Especially in safety-critical applications as automated driving, explainability and transferability are key to success. This motivates the use of model-based offline RL approaches, which leverage planning. However, current state-of-the art methods often neglect the influence of aleatoric uncertainty arising from the stochastic behavior of multi-agent systems. This work proposes a novel approach for Uncertainty-aware Model-Based Offline REinforcement Learning Leveraging plAnning (UMBRELLA), which solves the prediction, planning, and control problem of the SDV jointly in an interpretable learning-based fashion. A trained action-conditioned stochastic dynamics model captures distinctively different future evolutions of the traffic scene. The analysis provides empirical evidence for the effectiveness of our approach in challenging automated driving simulations and based on a real-world public dataset.