Embedding input uncertainties in hydrological modelling naturally leads to stochastic models, which render parameter calibration an often computationally intractable problem. We use a case study from urban hydrology based on a stochastic rain model, and we employ a highly efficient Hamiltonian Monte Carlo inference algorithm with a timescale separation to demonstrate that fully fledged Bayesian inference with stochastic models is no longer off-limits for hydrological applications.