We present a rapid timing analysis of optical (HiPERCAM and ULTRACAM) and X-ray (NICER) observations of the X-ray transient Swift J1858.6-0814 during 2018 and 2019. The optical light curves show relatively slow, large amplitude (~1 mag in gs) 'blue' flares (i.e. stronger at shorter wavelengths) on time-scales of ~minutes as well as fast, small amplitude (~0.1 mag in gs) 'red' flares (i.e. stronger at longer wavelengths) on time-scales of ~seconds. The 'blue' and 'red' flares are consistent with X-ray reprocessing and optically thin synchrotron emission, respectively, similar to what is observed in other X-ray binaries. The simultaneous optical versus soft- and hard-band X-ray light curves show time- and energy-dependent correlations. The 2019 March 4 and parts of the June data show a nearly symmetric positive cross-correlations (CCFs) at positive lags consistent with simple X-ray disc reprocessing. The soft- and hard-band CCFs are similar and can be reproduced if disc reprocessing dominates in the optical and one component (disc or synchrotron Comptonization) dominates both the soft and hard X-rays. A part of the 2019 June data shows a very different CCFs. The observed positive correlation at negative lag in the soft band can be reproduced if the optical synchrotron emission is correlated with the hot flow X-ray emission. The observed timing properties are in qualitative agreement with the hybrid inner hot accretion flow model, where the relative role of the different X-ray and optical components that vary during the course of the outburst, as well as on shorter time-scales, govern the shape of the optical/X-ray CCFs.