The conversion of ultrafast spin and orbital currents into charge currents underpins the generation of terahertz (THz) radiation in magnetic multilayers, yet the fundamental timescales of these processes remain largely unexplored. Here, we directly measure the absolute onset times from femtosecond laser excitation to pulsed THz emission in Fe/Pt, Co/Pt, and Ni/Pt bilayers grown on nonlinear crystals serving as temporal fiducials. We observe marked differences among the materials, with onset times of 95 fs, 21 fs, and 85 fs, respectively, revealing that the choice of ferromagnetic layer dictates distinct conduction current generation mechanisms dominated by spin or orbital dynamics. These results provide a direct time-resolved insight into spintronic and orbitronic THz emitters, establishing a quantitative benchmark for ultrafast spin-to-charge conversion in nanoscale ferromagnetic/nonmagnetic heterostructures.