Context. Submillimetre galaxies (SMGs) represent an important source population in the origin and cosmic evolution of the most massive galaxies. Hence, it is imperative to place firm constraints on the fundamental physical properties of large samples of SMGs. Aims. We determine the physical properties of a sample of SMGs in the COSMOS field that were pre- selected at the observed-frame wavelength of λobs = 1.1 mm, and followed up at λobs = 1.3 mm with the Atacama Large Millimetre/submillimetre Array (ALMA). Methods. We used the MAGPHYS model package to fit the panchromatic (ultraviolet to radio) spectral energy distributions (SEDs) of 124 of the target SMGs, which lie at a median redshift of z = 2.30 (19.4% are spectroscopically confirmed). The SED analysis was complemented by estimating the gas masses of the SMGs by using the λobs = 1.3 mm dust emission as a tracer of the molecular gas component. Results. The sample median and 16th– 84th percentile ranges of the stellar masses, obscured star formation rates, dust temperatures, and dust and gas masses were derived to be log(M⋆/M⊙) = 11.09+0.41-0.53, SFR = 402+661-233 M⊙ yr-1, Tdust = 39.7+9.7-7.4 K, log(Mdust/M⊙) = 9.01+0.20-0.31, and log(Mgas/M⊙ = 11.34+0.20- 0.23, respectively. The Mdust/M⋆ ratio was found to decrease as a function of redshift, while the Mgas/Mdust ratio shows the opposite, positive correlation with redshift. The derived median gas- to-dust ratio of 120+73-30 agrees well with the canonical expectation. The gas fraction (Mgas/ (Mgas + M⋆)) was found to range from 0.10 to 0.98 with a median of 0.62+0.27-0.23. We found that 57.3% of our SMGs populate the main sequence (MS) of star-forming galaxies, while 41.9% of the sources lie above the MS by a factor of greater than three (one source lies below the MS). These super-MS objects, or starbursts, are preferentially found at z ≳ 3, which likely reflects the sensitivity limit of our source selection. We estimated that the median gas consumption timescale for our SMGs is ~535 Myr, and the super-MS sources appear to consume their gas reservoir faster than their MS counterparts. We found no obvious stellar mass–size correlations for our SMGs, where the sizes were measured in the observed-frame 3 GHz radio emission and rest-frame UV. However, the largest 3 GHz radio sizes are found among the MS sources. Those SMGs that appear irregular in the rest-frame UV are predominantly starbursts, while the MS SMGs are mostly disk- like. Conclusions. The physical parameter distributions of our SMGs and those of the equally bright, 870 μm selected SMGs in the ECDFS field (the so-called ALESS SMGs) are unlikely to be drawn from common parent distributions. This might reflect the difference in the pre-selection wavelength. Albeit being partly a selection bias, the abrupt jump in specific SFR and the offset from the MS of our SMGs at z ≳ 3 might also reflect a more efficient accretion from the cosmic gas streams, higher incidence of gas-rich major mergers, or higher star formation efficiency at z ≳ 3. We found a rather flat average trend between the SFR and dust mass, but a positive SFR−Mgas correlation. However, to address the questions of which star formation law(s) our SMGs follow, and how they compare with the Kennicutt-Schmidt law, the dust- emitting sizes of our sources need to be measured. Nonetheless, the larger radio-emitting sizes of the MS SMGs compared to starbursts is a likely indication of their more widespread, less intense star formation activity. The irregular rest-frame UV morphologies of the starburst SMGs are likely to echo their merger nature. The current stellar mass content of the studied SMGs is very high, so they must quench to form the so- called red-and- dead massive ellipticals. Our results suggest that the transition from high-z SMGs to local ellipticals via compact, quiescent galaxies (cQGs) at z ~ 2 might not be universal, and the latter population might also descend from the so-called blue nuggets. However, z ≳ 4 SMGs could be the progenitors of higher redshift, z ≳ 3 cQGs, while our results are also consistent with the possibility that ultra-massive early-type galaxies found at 1.2 ≲ z ≲ 2 experienced an SMG phase at z ≤ 3.