We analyse the multi-frequency radio spectral properties of 41 6 GHz-detected ALMA-identified, submillimetre galaxies (SMGs), observed at 610 MHz, 1.4 GHz, and 6 GHz with GMRT and the VLA. Combining high-resolution (∼ 0.5 ′′ ) 6 GHz radio and ALMA 870 µm imaging (tracing rest-frame ∼ 20 GHz, and ∼ 250 µm dust continuum), we study the far-infrared/radio correlation via the logarithmic flux ratio q IR , measuring q IR = 2.20 ± 0.06 for our sample. We show that the high-frequency radio sizes of SMGs are ∼ 1.9 ± 0.4× (∼ 2-3 kpc) larger than those of the cool dust emission, and find evidence for a subset of our sources being extended on ∼ 10 kpc scales at 1.4 GHz. By combining radio flux densities measured at three frequencies, we can move beyond simple linear fits to the radio spectra of high-redshift star-forming galaxies, and search for spectral curvature, which has been observed in local starburst galaxies. At least a quarter (10/41) of our sample show evidence of a spectral break, with a median α 1.4 GHz 610 GHz = −0.60 ± 0.06, but α 6 GHz 1.4 GHz = −1.06 ± 0.04 obtained via stacking -a high-frequency flux deficit relative to simple extrapolations from the low-frequency data. We explore this result within this subset of sources in the context of age-related synchrotron losses, showing that a combination of weak magnetic fields (B ∼ 35 µG) and young ages (t SB ∼ 40-80 Myr) for the central starburst can reproduce the observed spectral break. Assuming these represent evolved (but ongoing) starbursts and we are observing these systems roughly half-way through their current episode of star formation, this implies starburst durations of 100 Myr, in reasonable agreement with estimates derived via gas depletion timescales.