We present single-dish 12CO(1-0) and 12CO(2-1) observations for 14 low-redshift quasi-stellar objects (QSOs). In combination with optical integral field spectroscopy, we study how the cold gas content relates to the star formation rate (SFR) and black hole accretion rate. 12CO(1-0) is detected in 8 of 14 targets and 12CO(2-1) is detected in 7 out of 11 cases. The majority of disc-dominated QSOs reveal gas fractions and depletion times matching normal star-forming systems. Two gas-rich major mergers show clear starburst signatures with higher than average gas fractions and shorter depletion times. Bulge-dominated QSO hosts are mainly undetected in 12CO(1-0), which corresponds, on average, to lower gas fractions than in disc-dominated counterparts. Their SFRs, however, imply shorter than average depletion times and higher star formation efficiencies. Negative QSO feedback through removal of cold gas seems to play a negligible role in our sample. We find a trend between black hole accretion rate and total molecular gas content for disc-dominated QSOs when combined with literature samples. We interpret this as an upper envelope for the nuclear activity and it is well represented by a scaling relation between the total and circumnuclear gas reservoir accessible for accretion. Bulge-dominated QSOs significantly differ from that scaling relation and appear uncorrelated with the total molecular gas content. This could be explained either by a more compact gas reservoir, blown out of the gas envelope through outflows, or a different interstellar medium phase composition.