In order to clarify the temporal interplay of the different photocurrent mechanisms occurring in single GaAs nanowire based circuits, we introduce an on-chip photocurrent pump-probe spectroscopy with a picosecond time resolution. We identify photoinduced thermoelectric, displacement, and carrier lifetime limited currents as well as the transport of photogenerated holes to the electrodes. Moreover, we show that the time-resolved photocurrent spectroscopy can be used to investigate the drift velocity of photogenerated carriers in semiconducting nanowires. Hereby, our results are relevant for nanowire-based optoelectronic and photovoltaic applications.