We present ALMA observations of the mid-J 12CO emission from six single-dish selected 870-μm sources in the Extended Chandra Deep Field-South and UKIDSS Ultra-Deep Survey fields. These six single-dish submillimetre sources were selected based on previous ALMA continuum observations, which showed that each comprised a blend of emission from two or more individual submillimetre galaxies (SMGs), separated on 5-10 arcsec scales. The six single-dish submillimetre sources targeted correspond to a total of 14 individual SMGs, of which seven have previously measured robust optical/near-infrared spectroscopic redshifts, which were used to tune our ALMA observations. We detect CO(3-2) or CO(4-3) at z = 2.3-3.7 in 7 of the 14 SMGs, and in addition serendipitously detect line emission from three gas-rich companion galaxies, as well as identify four new 3.3 mm selected continuum sources in the six fields. Joint analysis of our CO spectroscopy and existing data suggests that 64(± 18){ per cent} of the SMGs in blended submillimetre sources are unlikely to be physically associated. However, three of the SMG fields (50 per cent) contain new, serendipitously detected CO-emitting (but submillimetre-faint) sources at similar redshifts to the 870 μm selected SMGs we targeted. These data suggest that the SMGs inhabit overdense regions, but that these are not sufficiently overdense on ˜100 kpc scales to influence the source blending given the short lifetimes of SMGs. We find that 21± 12{ per cent} of SMGs have spatially distinct and kinematically close companion galaxies (˜8-150 kpc and ≲ 300 km s-1), which may have enhanced their star formation via gravitational interactions.