Subglacial drainage models represent water flow beneath glaciers and ice sheets. Here, we train fast statistical models called Gaussian process (GP) emulators to accelerate subglacial drainage modelling by ~ 1000 times. We use the fast emulator predictions to show that three of the model parameters are responsible for > 90 % of the variance in model outputs. The fast GP emulators will enable future uncertainty quantification and calibration of these models.