We investigate the radio properties of a sample of 850 μm-selected sources from the SCUBA-2 Cosmology Legacy Survey (S2CLS) using new deep, low-frequency radio imaging of the Lockman Hole field from the Low Frequency Array (LOFAR). Combining these data with additional observations at 324 MHz, 610 MHz, and 1.4 GHz from the Giant Metrewave Radio Telescope and the Jansky Very Large Array, we find a variety of radio spectral shapes and luminosities within our sample despite their similarly bright submillimetre flux densities. We characterise their spectral shapes in terms of multi-band radio spectral indices. Finding strong spectral flattening at low frequencies in ~20% of sources, we investigate the differences between sources with extremely flat low-frequency spectra and those with `normal' radio spectral indices (α > ‒0.25). As there are no other statistically significant differences between the two subgroups of our sample as split by the radio spectral index, we suggest that any differences are undetectable in galaxy-averaged properties that we can observe with our unresolved images, and likely relate to galaxy properties that we cannot resolve, on scales ≲1 kpc. We attribute the observed spectral flattening in the radio to free-free absorption, proposing that those sources with significant low-frequency spectral flattening have a clumpy distribution of star-forming gas. We estimate an average spatial extent of absorbing material of at most several hundred parsecs to produce the levels of absorption observed in the radio spectra. This estimate is consistent with the highest-resolution observations of submillimetre galaxies in the literature, which find examples of non-uniform dust distributions on scales of ~100 pc, with evidence for clumps and knots in the interstellar medium.