We study nonlinear effects in two-dimensional photonic metasurfaces supporting topologically protected helical edge states at the nanoscale. We observe strong third-harmonic generation mediated by optical nonlinearities boosted by multipolar Mie resonances of silicon nanoparticles. Variation of the pump-beam wavelength enables independent high-contrast imaging of either bulk modes or spin-momentum-locked edge states. We demonstrate topology-driven tunable localization of the generated harmonic fields and map the pseudospin-dependent unidirectional waveguiding of the edge states bypassing sharp corners. Our observations establish dielectric metasurfaces as a promising platform for the robust generation and transport of photons in topological photonic nanostructures.