We present a survey of the molecular gas in 61 submillimetre galaxies (SMGs) selected from 870$mu$m continuum surveys of the COSMOS, UDS and ECDFS fields, using the Atacama Large Millimeter Array (ALMA) and the Northern Extended Millimeter Array (NOEMA). 46 $^{12}$CO ($J=$2-5) emission lines are detected in 45 of the targets at $z=$1.2-4.8, with redshifts indicating that those which are submillimetre bright and undetected/faint in the optical/near-infrared typically lie at higher redshifts, with a gradient of $Delta z/Delta S_{870}=$0.11$pm$0.04mJy$^{-1}$. We also supplement our data with literature sources to construct a statistical CO spectral line energy distribution and find the $^{12}$CO line luminosities in SMGs peak at $J_{rm up}sim$6, consistent with the Cosmic Eyelash, among similar studies. Our SMGs lie mostly on or just above the main sequence, displaying a decrease in their gas depletion timescales $t_{rm dep} = M_{rm gas}/{rm SFR}$ with redshift in the range $zsim$1-5 and a median of 200$pm$50Myr at $zsim$2.8. This coincides with an increase in molecular gas fraction $mu_{rm gas} = M_{rm gas}/M_ast$ across the same redshift range. Finally we demonstrate that the $M_{rm baryon}$-$sigma$ distribution of 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}sim$5mJy may correspond to the division between slow- and fast-rotators seen in local early-type galaxies.
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