We present a study of the radio properties of 870$μ$m-selected submillimetre galaxies (SMGs), observed at high resolution with ALMA in the Extended Chandra Deep Field South. From our initial sample of 76 ALMA SMGs, we detect 52 SMGs at $>3σ$ significance in VLA 1400MHz imaging, of which 35 are also detected at $>3σ$ in new 610MHz GMRT imaging. Within this sample of radio-detected SMGs, we measure a median radio spectral index $α_{610}^{1400} = -0.79 ± 0.06$, (with inter-quartile range $α=[-1.16,-0.56]$) and investigate the far-infrared/radio correlation via the parameter $q_{\rm IR}$, the logarithmic ratio of the rest-frame 8-1000$μ$m flux and monochromatic radio flux. Our median $q_{\rm IR} = 2.56 ± 0.05$ (inter-quartile range $q_{\rm IR}=[2.42,2.78]$) is higher than that typically seen in single-dish 870$μ$m-selected sources ($q_{\rm IR} ∼ 2.4$), which may reflect the fact that our ALMA-based study is not biased to radio-bright counterparts, as previous samples were. Finally, we search for evidence that $q_{\rm IR}$ and $α$ evolve with age in a co-dependent manner, as predicted by starburst models: the data populate the predicted region of parameter space, with the stellar mass tending to increase along tracks of $q_{\rm IR}$ versus $α$ in the direction expected, providing the first observational evidence in support of these models. ; Comment: 13 pages, 4 figures, 3 colour figures, Accepted by MNRAS