Context. Young (≲600 Myr) low-mass stars (M ≲ 1 M_⊙) of equal mass exhibit a distribution of rotation periods. At the very early phases of stellar evolution, this distribution is set by the star-disc locking mechanism, which forces stars to rotate at the same rate as the inner edge of the disc. The primordial disc lifetime and consequently the duration of the disc-locking mechanism, can be significantly shortened by the presence of a close companion, making the rotation period distribution of close binaries different from that of either single stars or wide binaries. Aims. We use new data to investigate and better constrain the range of ages, the components separation, and the mass ratio dependence at which the rotation period distribution has been significantly affected by the disc dispersal that is enhanced by close companions. Methods. We select a sample of close binaries in the Upper Scorpius association (age ∼8 Myr) whose components have measured the separation and the rotation periods and compare their period distribution with that of coeval stars that are single stars. Results. We find that components of close binaries have, on average, rotation periods that are shorter than those of single stars. More precisely, binaries with approximately equal-mass components (0.9 ≤ M₂/M₁ ≤ 1.0) have rotation periods that are shorter than those of single stars by ∼0.4 d on average; the primary and secondary components of binaries with smaller mass ratios (0.8 < M₂/M₁ < 0.9) have rotation periods that are shorter than those of single stars by ∼1.9 d and ∼1.0 d on average, respectively. A comparison with the older 25 Myr β Pictoris association shows that whereas in the latter, all close binaries with projected separation ρ ≤ 80 AU rotate faster than single stars, in the Upper Scorpius this is only the case for about 70% of stars. Conclusions. We interpret the enhanced rotation in close binaries with respect to single stars as the consequence of an early disc dispersal induced by the presence of close companions. The enhanced rotation suggests that disc dispersal timescales are longest for single stars and shorter for close binaries.