We investigate, via time-resolved photoluminescence, the temperature-dependence of charge carrier recombination mechanisms in nonpolar (Al,Ga)N/GaN single quantum wells (QWs) grown via molecular beam epitaxy on the a-facet of bulk GaN crystals. We study the influence of both QW width and barrier Al content on the dynamics of excitons in the 10-320 K range. We first show that the effective lifetime of QW excitons s increases with temperature, which is evidence that nonradiative mechanisms do not play any significant role in the low-temperature range. The temperature range for increasing s depends on the QW width and Al content in the (Al,Ga)N barriers. For higher temperatures, we observe a reduction in the QW emission lifetime combined with an increase in the decay time for excitons in the barriers, until both exciton populations get fully thermalized. Based on analysis of the ratio between barrier and QW emission intensities, we demonstrate that the main mechanism limiting the radiative efficiency in our set of samples is related to nonradiative recombination in the (Al,Ga)N barriers of charge carriers that have been thermally emitted from the QWs.