Evaluation of the electron-nuclear dynamics and relaxation mechanisms of gold and silver nanoclusters and their alloys is important for future photocatalytic, light harvesting, and photoluminescence applications of these systems. In this work, the effect of silver doping on the nonradiative excited state relaxation dynamics of the atomically precise thiolate-protected gold nanocluster [AuAg(SH)] (n = 1, 12, 25) is studied theoretically. Time-dependent density functional theory is used to study excited states lying in the energy range 0.0-2.5 eV. The fewest switches surface hopping method with decoherence correction was used to investigate the dynamics of these states. The HOMO-LUMO gap increases significantly upon doping of 12 silver atoms but decreases for the pure silver nanocluster. Doped clusters show a different response for ground state population increase lifetimes and excited state population decay times in comparison to the undoped system. The ground state recovery times of the S-S states in the first excited peak were found to be longer for [AuAg(SH)] than the corresponding recovery times of other studied nanoclusters, suggesting that this partially doped nanocluster is best for preserving electrons in an excited state. The decay time constants were in the range of 2.0-20 ps for the six lowest energy excited states. Among the higher excited states, S has the slowest decay time constant although it occurs more quickly than S decay. Overall, these clusters follow common decay time constant trends and relaxation mechanisms due to the similarities in their electronic structures.