Accelerating mountain glacier recession in a warming climate threatens the sustainability of mountain water resources. The extent to which groundwater will provide resilience to these water resources is unknown, in part due to a lack of data and poorly understood interactions between groundwater and surface water. Here we address this knowledge gap by linking climate, glaciers, surface water, and groundwater into an integrated model of the Shullcas Watershed, Peru, in the tropical Andes, the region experiencing the most rapid mountain-glacier retreat on Earth. For a range of climate scenarios, our model projects that glaciers will disappear by 2100. The loss of glacial meltwater will be buffered by relatively consistent groundwater discharge, which only receives minor recharge (~2%) from glacier melt. However, increasing temperature and associated evapotranspiration, alongside potential decreases in precipitation, will decrease groundwater recharge and streamflow, particularly for the RCP 8.5 emission scenario.