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 (λrest = 1.3 mm). After subtracting the foreground galaxy contribution to the photometry, spectral energy distribution fitting yielded stellar masses of 109 − 11 M⊙, with star formation rates of 25−160 M⊙ yr−1 for the host galaxies. These QSOs have lower LCO′ 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(αCO/4) Myr. We find molecular gas masses in the range ≤2−40 × 109(αCO/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.