Flood extent maps derived from satellite imagery are increasingly important for flood response and risk management. Although recent deep learning models have substantially improved segmentation accuracy, they typically provide deterministic predictions without indicating their reliability. This limits their operational value in regions affected by cloud contamination, spectrally ambiguous floodwater, or environmental conditions insufficiently represented in the training data. This study develops an uncertainty-aware flood mapping framework by integrating Evidential Deep Learning (EDL) into an operational cloud-aware mapping workflow. The proposed framework simultaneously generates flood extent and pixel-level predictive uncertainty, enabling uncertainty information to be incorporated directly into the mapping process. Its performance is evaluated against a deterministic UNet++ baseline and two established uncertainty estimation approaches, Deep Ensemble and MC Dropout, using the extended WorldFloods dataset. Across the test dataset, the three uncertainty-aware methods achieved comparable IoU values, all consistently outperforming the deterministic baseline, indicating that predictive uncertainty can be incorporated without degrading segmentation performance. Their uncertainty behaviour and computational requirements, however, differed substantially. Among the evaluated approaches, the evidential formulation provided the most favourable operational trade-off by preserving segmentation performance, producing stable epistemic uncertainty estimates across diverse flood events, and requiring only a single model evaluation per image tile. These findings demonstrate that uncertainty quality should be regarded as a complementary evaluation criterion alongside segmentation accuracy. By combining competitive segmentation performance, informative uncertainty estimates, and computational efficiency, the proposed framework provides a practical framework for operational uncertainty-aware satellite-based flood mapping.