Afterglows of gamma-ray bursts (GRBs) from Population III (Pop III) stars could reveal the formation history and properties of these first generation stars. Through detailed simulation, we predict the prospects of detecting these afterglows with a range of established, existing and upcoming telescopes across the spectrum from radio waves to X-rays. The simulations show that the afterglow light curves of Pop III GRBs at high redshift (≳8) are very similar to those of Pop I/II GRBs at lower redshift (∼2), with the distinction that Lyα absorption at Pop III redshifts removes any optical [and some near-infrared (NIR)] component. We calculate that within a single field of view (FOV) of the Australian Square Kilometre Array Pathfinder (ASKAP) telescope there will be on average four detectable Pop III GRB afterglows. This is the product of ASKAP's large FOV and excellent sensitivity at wavelengths where the afterglows are very long-lasting. We show that the exceptional sensitivity of the James Webb Space Telescope (JWST) Near-InfraRed Camera will make this the optimal instrument for afterglow follow-up and redshift measurement, while JWST Near-InfraRed Spectrograph will be able to detect the absorption features of Pop III-enriched environments in 70 per cent of directed Pop III GRB afterglows. We also find that the Atacama Large Millimetre Array is very poorly suited to observe these afterglows, and that the Spectrum-Roentgen-Gamma 4 yr all-sky X-ray survey has a 12 per cent chance of detecting an orphan Pop III GRB afterglow. The optimal strategy for detecting, identifying and studying Pop III GRB afterglows is to have JWST attempt NIR photometry of afterglows with a detected radio component but no detected optical component.