Effective river management relies heavily on the accurate simulation of river flow dynamics to develop scenario-based strategies and inform decision making. Deep learning (DL) models, as alternatives to conventional simulation techniques, offer a compelling blend of precision and efficiency. This study introduces FNOCL (Fourier Neural Operator with ConvLSTM units), a novel DL neural network for modelling unsteady natural river flows. The network is adept at handling the complexities of spatial and temporal variations. FNOCL builds upon the Fourier neural operator (FNO) by incorporating convolutional long short-term memory (ConvLSTM) layers, combining robust performance with high proficiency in terms of capturing spatiotemporal patterns. Notably, FNOCL accommodates complex river cross-section profiles, thus enabling the accurate modelling of flow dynamics within intricate river morphologies. Our study applies FNOCL to two distinct river networks with varying morphology complexity levels. With reference to the HEC-RAS model, FNOCL showcases its ability to attain high accuracy in model prediction tasks and superior generalization with regard to addressing the uncertainty associated with inflow variations and cross-sectional changes. In comparative analyses with benchmark models (i.e., FNO and ConvLSTM), FNOCL performs most efficiently during the training process and obtains the most accurate water level and river flow predictions in the spatiotemporal domain. Furthermore, FNOCL exhibits a substantial speedup over HEC-RAS, notably reaching up to 50 times in a complex river network. These results highlight FNOCL as an efficient surrogate for conventional simulators, facilitating precise analyses across various scenarios. It provides efficient computations for system responses within complex river flow networks, thereby enhancing applications in river management, such as flood risk assessment, flood control, and the planning and design of hydraulic control structures.