2025MNRAS.539.1908O

Binary neutron star mergers and collapsing massive stars can both create millisecond magnetars. Such magnetars are candidate engines to power gamma-ray bursts (GRBs). The non-thermal light curve of the resulting transients can exhibit multiple components, including the GRB afterglow, pulsar wind nebula (PWN), and ejecta afterglow. We derive the time-scales for the peak of each component and show that the PWN is detectable at radio frequencies, dominating the emission for $\sim$6 yr for supernova/long GRBs (SN/LGRBs) and $\sim$ 100 d for kilonova/short GRBs (KN/SGRBs) at 1 GHz, and $\sim$1 yr for SN/LGRBs and $\sim$ 15 d for KN/SGRBs at 100 GHz. The PWN emission has an exponential, frequency-dependent rise to peak that cannot be replicated by an ejecta afterglow. We show that PWNe in SN/LGRBs can be detected out to $z \sim 0.06$ with current instruments and $z \sim 0.3$ with next-generation instruments and PWNe in KN/SGRBs can be detected out to $z \sim 0.3$ with current instruments and $z \sim 1.5$ with next-generation instruments. We find that the optimal strategy for detecting PWNe in these systems is a multiband, high cadence radio follow-up of nearby KN/SGRBs with an X-ray plateau or extended prompt emission from 10 to 100 d post-burst.