In this study, we elaborate on and evaluate a new reduced basis method for model reduction of the shallow water equations using Proper Orthogonal Decomposition (POD) and artificial Neural Networks (NNs). The method begins with the POD technique to construct reduced bases from high-resolution solutions, followed by training two deep NNs to learn associated coefficients in the reduced bases. The approach follows an offline-online strategy: the POD reduced basis, along with the training of the NNs, is performed in an offline stage, and then the surrogate model can be used in an online stage for real-time predictions. The method takes into account the POD-based projection error, enabling the attainment of higher accuracy while preserving a limited number of POD modes, even in the delicate situation of convection-dominated flow problems. This point is crucial in our approach since it enables to limit the output dimension of the NNs, thus providing the opportunity to employ smaller NNs (with less parameters), and therefore smaller dataset (the snapshots) to train the NNs. The process is non-intrusive: it does not require opening the highresolution model code. The method is evaluated on a real-world test case aimed at simulating inundation of the Aude river (Southern France). The results show that the proposed method provides satisfying accuracy for the hydraulic variables (water elevation, discharge) compared to the reference high-resolution 2D shallow water model. Overall, the method is promising, particularly for performing real-time simulations of flood plain dynamics.