We report on the detection of optical/near-infrared (O-IR) quasi-periodic oscillations (QPOs) from the black hole (BH) X-ray transient Swift J1727.8–1613. We obtained three X-ray and O-IR high-time-resolution observations of the source during its intermediate state (2023 September 9, 15, and 17) using NICER, HAWK-I@VLT, HIPERCAM@GTC, and ULTRACAM@NTT. We clearly detected a QPO in the X-ray and O-IR bands during all three epochs. The QPO evolved, drifting from 1.4 Hz in the first epoch, up to 2.2 Hz in the second, and finally reaching 4.2 Hz in the third epoch. These are among the highest O-IR QPO frequencies detected for a BH X-ray transient. During the first two epochs, the X-ray and O-IR emission are correlated, with an optical lag (compared to the X-rays) varying from +70 to 0 ms. Finally, during the third epoch, we measured, for the first time, a lag of the $z_s$ band with respect to the $g_s$ band at the QPO frequency ($\approx$ +10 ms). By estimating the variable O-IR SED we find that the emission is most likely non-thermal. Current state-of-the-art models can explain some of these properties, but neither the jet nor the hot flow model can easily explain the observed evolution of the QPOs. While this allowed us to put tight constraints on these components, more frequent coverage of the state transition with fast multiwavelength observations is still needed to fully understand the evolution of the disc/jet properties in BH low-mass X-ray binaries.