We report terahertz (THz)-pump/mid-infrared probe near-field studies on Si-doped GaAs-InGaAs core-shell nanowires utilizing THz radiation from the free-electron laser FELBE. Upon THz excitation of free carriers, we observe a red shift of the plasma resonance in both amplitude and phase spectra, which we attribute to the heating of electrons in the conduction band. The simulation of heated electron distributions anticipates a significant electron population in both the L- and X-valleys. The two-temperature model is utilized for quantitative analysis of the dynamics of the electron gas temperature under THz pumping at various power levels.