Exciton dynamics in dilute nitride GaP/GaNP core/shell nanowires (NWs) with pronounced band-tail states formed by nitrogen clusters is investigated using time-resolved photoluminescence (PL) spectroscopy. The emission of excitons localized at the N-related states in the GaNP shell is found to exhibit a stretched exponential decay, with the 1/e lifetime dramatically shortened with decreasing excitation wavelength and reduced shell thickness. The observed PL transient behavior is explained by markedly different exciton lifetimes between the surface and bulk regions of the GaNP shell, that is, similar to 20 ps versus similar to 10 ns, respectively. Despite being trapped at the deep localized N states, the photoexcited excitons are concluded to suffer from pronounced surface recombination via tunneling to the surface states within a distance of 10 nm from the surface, which results in the depth-dependent PL dynamics. The surface recombination rate is, however, lower than that previously reported for GaP, indicative of partial passivation of the surface states by nitrogen. From temperature-dependent PL measurements, characteristic thermal activation energies for the surface and bulk-related nonradiative recombination channels are deduced. The obtained results provide insight into the exciton/carrier dynamics in NW systems with strong localization or alloy disorder, which is important for future nanophotonic and photovoltaic applications of such structures.