Submillimetre galaxies (SMGs) in the early Universe are potential antecedents of the most massive galaxies we see in the present-day Universe. An important step towards quantifying this galactic evolutionary connection is to investigate the fundamental physical properties of SMGs, such as their stellar mass content (M⋆) and star formation rate (SFR). We attempt to characterise the physical nature of a 1.1 mm selected, flux-limited, and interferometrically followed up sample of SMGs in the COSMOS field. Methods. We used the latest release of the MAGPHYS code to fit the multiwavelength (UV to radio) spectral energy distributions (SEDs) of 16 of the target SMGs, which lie at redshifts z ≃ 1.6−5.3. We also constructed the pure radio SEDs of our SMGs using three different radio bands (325 MHz, 1.4 GHz, and 3 GHz). Moreover, since two SMGs in our sample, AzTEC 1 and AzTEC 3, benefit from previous ^(12)C^(16)O line observations, we studied their properties in more detail. The median and 16th–84th percentile ranges of M⋆, infrared (8−1000 μm) luminosity (L_(IR)), SFR, dust temperature (T_(dust)), and dust mass (M_(dust)) were derived to be log (M⋆/M⊙) = 10.96^(+0.34)_(-0.19), log (L_(IR)/L⊙) = 12.93^(+0.09)_(-0.19), SFR = 856^(+191)_(-310)M⊙ yr^(-1), T_(dust) = 40.6^(+7.5)_(-8.1) K, and log (M_(dust)/M⊙) = 9.17^(+0.03)_(-0.33), respectively. We found that 63% of our target SMGs lie above the galaxy main sequence by more than a factor of 3 and, hence, are starbursts. The 3 GHz radio sizes we have previously measured for the target SMGs were compared with the present M⋆ estimates, and we found that the z > 3 SMGs are fairly consistent with the mass–size relationship of z ~ 2 compact, quiescent galaxies (cQGs). The median radio spectral index is found to be α = −0.77^(+0.28)_(-0.42). The median IR-radio correlation parameter is found to be q = 2.27^(+0.27)_(-0.13), which is lower than was measured locally (median q = 2.64). The gas-to- dust mass ratio for AzTEC 1 is derived to be δ_(gdr) = 90^(+23)_(-19), while that for AzTEC 3 is 33^(+28)_(-18). AzTEC 1 is found to have a sub-Eddington SFR surface density (by a factor of 2.6^(+0.2)_(-0.1)), while AzTEC 3 appears to be an Eddington-limited starburster. The gas reservoir in these two high-z SMGs would be exhausted in only ~ 86 and 19 Myr at the current SFR, respectively. Conclusions. A comparison of the MAGPHYS-based properties of our SMGs with those of equally bright SMGs in the ECDFS field (the ALESS SMGs) that are 870 μm selected and followed up by ALMA, suggests that the two populations share fairly similar physical characteristics, including the q parameter. The somewhat higher L_(dust) for our sources (factor of 1.9^(+9.3)_(-1.6) on average) can originate in the longer selection wavelength of 1.1 mm. Although the derived median α is consistent with a canonical synchrotron spectral index, some of our SMGs exhibit spectral flattening or steepening, which can be attributed to different cosmic-ray energy gain and loss mechanisms. A hint of negative correlation is found between the 3 GHz size and the level of starburstiness and, hence, cosmic-ray electrons in more compact starbursts might be more susceptible to free-free absorption. Some of the derived low and high q values (compared to the local median) could be the result of a specific merger or post-starburst phase of galaxy evolution. Overall, our results, such as the M⋆–3 GHz radio size analysis and comparison with the stellar masses of z ~ 2 cQGs in concert with the star formation properties of AzTEC 1 and 3, support the scenario where z > 3 SMGs evolve into the present day giant, gas-poor ellipticals.