c-plane GaN/AlN quantum dots (QDs) are promising zero-dimensional quantum nanostructures that exhibit single photon emission properties up to room temperature and even above. In this context, it is of prime interest to gain a deeper insight into the recombination dynamics of photogenerated electron-hole pairs captured by such dots. Hence, in this work, we study the time-resolved photoluminescence (PL) properties in the low injection regime and at cryogenic temperatures of c-plane GaN/AlN QD ensembles emitting above the bulk GaN bandgap in order to properly understand the nature of the recombination channels behind the observed non-exponential decay time profiles. Such decays reveal the existence of a relaxation channel competing with the radiative recombination one. It is thus observed that for the former process the dynamics is independent of the dot height, which is attributed to a reversible nonradiative transfer that could be mediated by a spin-flip process to a dark-level state. The radiative recombination process is recognizable thanks to the characteristic dependence of its lifetime with the emission energy, which is well accounted for by the built-in electric field inherent to quantum nanostructures grown along the c axis and the variations in the lateral confinement at play in such QDs. Those conclusions are drawn from the analysis of the time evolution of the PL spectra by means of a simple analytical model that enables to exclude any screening of the built-in electric field.