Deep learning models have shown promising performance in the field of diabetic retinopathy (DR) staging. However, collaboratively training a DR staging model across multiple institutions remains a challenge due to non-iid data, client reliability, and confidence evaluation of the prediction. To address these issues, we propose a novel federated uncertainty-aware aggregation paradigm (FedUAA), which considers the reliability of each client and produces a confidence estimation for the DR staging. In our FedUAA, an aggregated encoder is shared by all clients for learning a global representation of fundus images, while a novel temperature-warmed uncertainty head (TWEU) is utilized for each client for local personalized staging criteria. Our TWEU employs an evidential deep layer to produce the uncertainty score with the DR staging results for client reliability evaluation. Furthermore, we developed a novel uncertainty-aware weighting module (UAW) to dynamically adjust the weights of model aggregation based on the uncertainty score distribution of each client. In our experiments, we collect five publicly available datasets from different institutions to conduct a dataset for federated DR staging to satisfy the real non-iid condition. The experimental results demonstrate that our FedUAA achieves better DR staging performance with higher reliability compared to other federated learning methods. Our proposed FedUAA paradigm effectively addresses the challenges of collaboratively training DR staging models across multiple institutions, and provides a robust and reliable solution for the deployment of DR diagnosis models in real-world clinical scenarios.