We present a study of the radio properties of 870 {mu}m-selected submillimetre galaxies (SMGs), observed at high resolution with Atacama Large Millimeter Array (ALMA) in the Extended Chandra Deep Field South. From our initial sample of 76 ALMA SMGs, we detect 52 SMGs at >3{sigma} significance in Karl G. Jansky Very Large Array 1400MHz imaging, of which 35 are also detected at >3{sigma} in new 610MHz Giant Metre-Wave Radio Telescope imaging. Within this sample of radio-detected SMGs, we measure a median radio spectral index {alpha}_{610}_^{1400}^=-0.79+/-0.06, (with inter-quartile range {alpha}=[-1.16,-0.56]) and investigate the far-infrared/radio correlation via the parameter q_IR_, the logarithmic ratio of the rest-frame 8-1000{mu}m flux and monochromatic radio flux. Our median q_IR_=2.56+/-0.05 (inter-quartile range q_IR_=[2.42,2.78]) is higher than that typically seen in single-dish 870{mu}m-selected sources (q_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_IR_ and {alpha} evolve with age in a codependent 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_IR_ versus {alpha} in the direction expected, providing the first observational evidence in support of these models.