This work presents a method to emulate the flow dynamics of physically based hydrodynamic simulators under variations of time-dependent rainfall and parametric scenarios. Although surrogate modelling is often employed to deal with the computational burden of this type of simulators, common techniques used for model emulation as polynomial expansions or Gaussian processes cannot deal with large parameter space dimensionality. This restricts their applicability to a reduced number of static parameters under a fixed rainfall process. The technique presented combines the use of a modified Unit Hydrograph (UH) scheme and a polynomial chaos expansion (PCE) to emulate flow from physically based hydrodynamic models. The novel element of the proposed methodology is that the emulator compensates for the errors induced by the assumptions of proportionality and superposition of the UH theory when dealing with non-linear model structures, whereas it approximates properly the behaviour of a physically based simulator to new (spatially-uniform) rainfall time-series and parametric scenarios. The computational time is significantly reduced, which makes the practical use of the model feasible (e.g. real time control, flood warning schemes, hydraulic structures design, parametric inference etc.). The applicability of this methodology is demonstrated in three case studies, through the emulation of a simplified non-linear tank-in-series routing structure and of the 2D Shallow Water Equations (2D-SWE) solution (FLOW-R2D) in two computational domains. Results indicate that the proposed emulator can approximate with a high degree of accuracy the behaviour of the original models under a wide range of rainfall inputs and parametric values.