We investigate the lasing dynamics of optically-pumped submicrowatt-threshold Raman silicon lasers that employ a high-quality (high-Q) nanocavity design. The measurements reveal that free carriers generated by two-photon absorption induce dynamic effects during the initial lasing process, even at the very low threshold power of 0.12 µW. At higher excitation powers, the Raman laser signal exhibits a significant reduction within a few hundred nanoseconds after the initial rise, followed by clear oscillations. We find that the temporal behavior of the laser signal strongly depends on the excitation wavelength. However, the Raman laser signal converges to a stable continuous operation within a few microseconds after the initial rise for any excitation power employed in this work. Numerical simulations indicate that the oscillations reflect the dynamic shift of the resonant wavelength due to the thermo-optic effect and the carrier-plasma effect, which are induced by free carriers generated via two-photon absorption. These results are useful for understanding the correct design of future devices that employ Raman silicon nanocavity lasers.