  We analyse the multi-frequency radio spectral properties of $41$
6GHz-detected ALMA-identified, submillimetre galaxies (SMGs), observed at
610MHz, 1.4GHz, 6GHz with GMRT and the VLA. Combining high-resolution
($\sim0.5''$) 6GHz radio and ALMA $870\,\mu$m imaging (tracing rest-frame
$\sim20$GHz, and $\sim250\,\mu$m dust continuum), we study the
far-infrared/radio correlation via the logarithmic flux ratio $q_{\rm IR}$,
measuring $\langle q_{\rm IR}\rangle=2.20\pm 0.06$ for our sample. We show that
the high-frequency radio sizes of SMGs are $\sim1.9\pm 0.4\times$
($\sim2$-$3$kpc) larger than those of the cool dust emission, and find evidence
for a subset of our sources being extended on $\sim 10$kpc scales at 1.4GHz. 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 $\langle\alpha^{1.4\,{\rm GHz}}_{610\,{\rm
GHz}}\rangle=-0.60\pm 0.06$, but $\langle\alpha^{6\,{\rm GHz}}_{1.4\,{\rm
GHz}}\rangle=-1.06\pm 0.04$ -- 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 ($B\sim35\,\mu$G) and young ages
($t_{\rm SB}\sim40$--$80\,$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
$\lesssim100$Myr, in reasonable agreement with estimates derived via gas
depletion timescales.
