We report on first THz-pump / MIR-probe SNOM studies on Si-doped GaAs-InGaAs core-shell NWs utilizing THz radiation from the free-electron laser FELBE. Upon intraband THz-pump we observe a red shift of the NW plasma resonance in both amplitude and phase spectra, while a controlled interband optical pumping induces a blue shift of the resonance. In both cases, the signal exponentially decays with a time constant of 4-5 ps. We attribute the blue shift to the contribution of photogenerated charge carriers, while the red shift is assigned to the heating of electrons in the conduction band accelerated by the THz electric field of the pump pulses and the subsequent increase of their effective mass due to the nonparabolic Γ-valley dispersion.