We have used ALMA and NOEMA to study the molecular gas reservoirs in 61 ALMA-identified submillimetre galaxies (SMGs) in the COSMOS, UDS, and ECDFS fields. We detect ^12^CO (J_up_=2-5) emission lines in 50 sources, and [CI](^3^P_1_-^3^P_0_) emission in eight, at z=1.2-4.8 and with a median redshift of 2.9+/-0.2. By supplementing our data with literature sources, we construct a statistical CO spectral line energy distribution and find that the ^12^CO line luminosities in SMGs peak at J_up_~6, consistent with similar studies. We also test the correlations of the CO, [CI], and dust as tracers of the gas mass, finding the three to correlate well, although the CO and dust mass as estimated from the 3-mm continuum are preferable. We estimate that SMGs lie mostly on or just above the star-forming main sequence, with a median gas depletion timescale, t_dep_=M_gas_/SFR, of 210+/-40Myr for our sample. Additionally, t_dep_ declines with redshift across z~1-5, while the molecular gas fraction, {mu}_gas_=M_gas_/M_*_, increases across the same redshift range. Finally, we demonstrate that the distribution of total baryonic mass and dynamical line width, M_baryon_-{sigma}, for our SMGs is consistent with that followed by early-type galaxies in the Coma cluster, providing strong support to the suggestion that SMGs are progenitors of massive local spheroidal galaxies. On the basis of this, we suggest that the SMG populations above and below an 870-{mu}m flux limit of S_870_~5mJy may correspond to the division between slow and fast rotators seen in local early-type galaxies.