Transition-metal dichalcogenide heterostructures are an emergent platform for novel many-body states from exciton condensates to nanolasers. However, their exciton dynamics are difficult to disentangle due to multiple competing processes with time scales varying over many orders of magnitude. Using a configurable nano-optical cavity based on a plasmonic scanning probe tip, the radiative (rad) and nonradiative (nrad) relaxation of intra- and interlayer excitons is controlled. Tuning their relative rates in a WSe2/MoSe2 heterobilayer over 6 orders of magnitude in tip-enhanced photoluminescence spectroscopy reveals a cavityinduced crossover from nonradiative quenching to Purcell-enhanced radiation. Rate equation modeling with the interlayer charge transfer time as a reference clock allows for a comprehensive determination from the long interlayer exciton (IX) radiative lifetime tau(rad)(IX) = (94 +/- 27) ns to the 5 orders of Irad magnitude faster competing nonradiative lifetime tau(rad)(IX) = (0.6 +/- 0.2) ps. This approach of nanocavity clock spectroscopy is generally applicable to a wide range of excitonic systems with competing decay pathways.