The traditional view of a dry, volatile-poor Moon has been challenged by the identification of water and other volatiles in lunar samples, but the volatile budget delivery time (s), source (s) and temporal evolution remain poorly constrained. Here we show that hydrogen and chlorine isotopic ratios in lunar apatite changed significantly during the Late Accretion (LA, 4.1-3.8 billion years ago). During this period, deuterium/hydrogen ratios in the Moon changed from initial carbonaceous-chondrite-like values to values consistent with an influx of ordinary-chondrite-like material and pre-LA elevated delta Cl-37 values drop towards lower chondrite-like values. Inferred pre-LA lunar interior water contents are significantly lower than pristine values suggesting degassing, followed by an increase during the LA. These trends are consistent with dynamic models of solar-system evolution, suggesting that the Moon's (and Earth's) initial volatiles were replenished similar to 0.5 Ga after their formation, with their final budgets reflecting a mixture of sources and delivery times.