Unobscured quasars (QSOs) are predicted to be the final stage in the evolutionary sequence from gas-rich mergers to gas-depleted, quenched galaxies. Studies of this population, however, find a high incidence of far-infrared-luminous sources-suggesting significant dust-obscured star formation-but direct observations of the cold molecular gas fuelling this star formation are still necessary. We present a NOEMA study of CO(2-1) emission, tracing the cold molecular gas, in ten lensed z = 1-1.5 unobscured QSOs. We detected CO(2-1) in seven of our targets, four of which also show continuum emission (λ_rest = 1.3 mm). After subtracting the foreground galaxy contribution to the photometry, spectral energy distribution fitting yielded stellar masses of 10^9-11 M_⊙, with star formation rates of 25-160 M_⊙ yr^-1 for the host galaxies. These QSOs have lower L′_CO than star-forming galaxies with the same LIR, and show depletion times spanning a large range (50-900 Myr), but with a median of just 90(α_CO/4) Myr. We find molecular gas masses in the range ≤2-40 × 10⁹(α_CO/4) M_⊙, which suggest gas fractions above ~50% for most of the targets. Despite the presence of an unobscured QSO, the host galaxies are able to retain significant amounts of cold gas. However, with a median depletion time of ~90 Myr, the intense burst of star formation taking place in these targets will quickly deplete their molecular gas reservoirs in the absence of gas replenishment, resulting in a quiescent host galaxy. The non-detected QSOs are three of the four radio-loud QSOs in the sample, and their properties indicate that they are likely already transitioning into quiescence. Recent cosmological simulations tend to overestimate the depletion times expected for these z ~ 1 QSO-host galaxies, which is likely linked to their difficulty producing starbursts across the general high-redshift galaxy population.