We present an experimental setup capable of time-resolved photoluminescence spectroscopy for photon energies in the range of 0.51 to 0.56 eV with an instrument time response of 75 ps. The detection system is based on optical parametric three-wave mixing, operates at room temperature, has spectral resolving power, and is shown to be well suited for investigating dynamical processes in germanium-tin alloys. In particular, the carrier lifetime of a direct-bandgap ${\mathrm{Ge}}_{1?x}{\mathrm{Sn}}_x$ film with concentration $x=12.5\%$ and biaxial strain $?0.55\%$ is determined to be $217\pm 15\mathrm{ps}$ at a temperature of 20 K. A room-temperature investigation indicates that the variation in this lifetime with temperature is very modest. The characteristics of the photoluminescence as a function of pump fluence are discussed.