Subglacial drainage networks regulate the response of ice sheet flow to surface meltwater input to the subglacial environment. Simulating subglacial hydrology evolution is critical to projecting ice sheet sensitivity to climate, and contribution to sea-level change. However, current numerical subglacial hydrology models are computationally expensive, and, consequently, evolving subglacial hydrology is neglected in large-scale ice sheet simulations. We present a deep learning emulator of a state-of-the-art subglacial hydrology model, trained at multiple Greenland glaciers. Our emulator performs strongly in both temporal (R2>0.99) and spatial (R2>0.96) generalization, offers high computational savings, and can be used to force numerical ice sheet models. This will enable century- and large-scale ice sheet model simulations, including interactions between ice flow and increased meltwater input to the subglacial environment. Generally, our work demonstrates that machine learning can further improve ice sheet models, reduce computational bottlenecks, and exploit information from high-fidelity models and novel observational platforms.