In semiconductor nanocrystals, excited electrons relax through multiple radiative and nonradiative pathways. This complexity complicates characterization of their decay processes with standard time- and temperature-dependent photoluminescence studies. Here, we exploit a simple nanophotonic approach to augment such measurements and to address open questions related to nanocrystal emission. We place nanocrystals at different distances from a gold reflector to affect radiative rates through variations in the local density of optical states. We apply this approach to spherical CdSe-based nanocrystals to probe the radiative efficiency and polarization properties of the lowest dark and bright excitons by analyzing temperature-dependent emission dynamics. For CdSe-based nanoplatelets, we identify the charge-carrier trapping mechanism responsible for strongly delayed emission. Our method, when combined with careful modeling of the influence of the nanophotonic environment on the relaxation dynamics, offers a versatile strategy to disentangle the complex excited-state decay pathways present in fluorescent nanocrystals as well as other emitters.