Nanowire lasers can be monolithically and site-selectively integrated onto silicon photonic circuits. To assess their full potential for ultrafast opto-electronic devices, a detailed understanding of their lasing dynamics is crucial. However, the roles played by their resonator geometry and the microscopic processes that mediate energy exchange between the photonic, electronic, and phononic systems are largely unexplored. Here, we apply femtosecond pump-probe spectroscopy to show that GaAs-AlGaAs core-shell nanowire lasers exhibit sustained intensity oscillations with frequencies ranging from 160 GHz to 260 GHz. These dynamics are intricately linked to the strong interaction between the lasing mode and the gain material arising from their wavelength-scale dimensions. Combined with dynamic competition between photoinduced carrier heating and cooling via phonon scattering, this enables self-induced electron-hole plasma temperature oscillations, which modulate the laser output. We anticipate that our results will lead to new approaches for ultrafast intensity and phase modulation of chip-integrated nanoscale semiconductor lasers.