We present high-cadence multifrequency radio observations of the long gamma-ray burst (GRB) 190829A, which was detected at photon energies above 100 GeV by the High Energy Stereoscopic System (H.E.S.S.). Observations with the Meer Karoo Array Telescope (MeerKAT, 1.3 GHz) and Arcminute Microkelvin Imager – Large Array (AMI-LA, 15.5 GHz) began one day post-burst and lasted nearly 200 d. We used complementary data from Swift X-Ray Telescope (XRT), which ran to 100 d post-burst. We detected a likely forward shock component with both MeerKAT and XRT up to over 100 d post-burst. Conversely, the AMI-LA light curve appears to be dominated by reverse shock emission until around 70 d post-burst when the afterglow flux drops below the level of the host galaxy. We also present previously unpublished observations of the other H.E.S.S.-detected GRB, GRB 180720B from AMI-LA, which shows likely forward shock emission that fades in less than 10 d. We present a comparison between the radio emission from the three GRBs with detected very high energy (VHE) gamma-ray emission and a sensitivity-limited radio afterglow sample. GRB 190829A has the lowest isotropic radio luminosity of any GRB in our sample, but the distribution of luminosities is otherwise consistent, as expected, with the VHE GRBs being drawn from the same parent distribution as the other radio-detected long GRBs.