In this work we employ photoluminescence (PL) transient spectroscopy to study exciton and free carrier recombination dynamics in novel GaAs/GaNAs core/shell NWs grown by plasma-assisted molecular beam epitaxy (MBE) on (111) plane Si substrates. It is shown that the PL emission in these structures is dominated by radiative recombination of localized excitons (LE) at T< 100 K and free exciton/carriers at T > 100K, both of which occur within the GaNAs shell. The LE dynamics is found to be determined by their energy transfer from shallow to deep localized states and also radiative recombination. The latter has a typical lifetime of around 5 ns and predominantly governs transient behavior of the deeply localized excitons. The fact that this radiative lifetime is insensitive to measurement temperature could suggest that the deeply localized states within the GaNAs alloy experience three-dimensional quantum confinement. On the other hand, PL decay arising from FE/free carriers that dominates at T>100K is found to be significantly faster and is governed by nonradiative recombination which occurs predominantly within the near-surface region of the GaNAs shell. The contribution of the surface recombination, which shortens the exciton lifetime down to tens of picoseconds, is concluded to be enhanced in the NW structures with a higher surface-to-volume ratio. This conclusion is based on the comparison of exciton dynamics in the NW structures with that in the reference GaNAs epilayer. Most importantly, the surface recombination becomes partially suppressed with increasing nitrogen composition in the alloy. This conclusion is based on our experimental finding that lifetimes of photo-generated free excitons and free carriers increase with increasing N composition - see Figure 1. This is accompanied by a sizable enhancement in the room temperature PL intensity in the GaAs/GaNAs core/shell NWs as compared with reference GaAs NW structures. The observed N-induced suppression of the surface recombination is suggested to be a result of an N-induced modification of the surface states that are responsible for the nonradiative recombination. Our results, therefore, demonstrate the great potential of incorporating GaNAs in III-V NWs to achieve efficient nano-scale light emitters grown on Si substates, which offers a number of attractive added values such as large freedom in band and lattice engineering as well as inherent passivation of harmful surface states.