We employ time-resolved photoluminescence combined with scanning electron microscopy and modeling to evaluate the lifetimes of excitons and free carriers in core-shell ZnTe-Zn1-xMgxTe nanowires. We find that electron tunneling through the shell to the surface controls the decay dynamics. The photoluminescence of single nanowires reveals contributions from an electron-hole plasma. The analysis of its temporal behavior allows one to extract the carrier and exciton lifetimes and monitor the cooling dynamics. In particular, we demonstrate that most of the electrons tunnel out before they cool down or bind into excitons. A semiclassical model allows us to extract the contributions of tunneling and recombination to the photoluminescence decay. We find that the recombination time shortens with increasing temperature as a result of an activation of a phonon-assisted nonradiative process.