The black hole binary, V404 Cygni, went into outburst in 2015 June, after 26 years of X-ray quiescence. We observed the outburst with the Neil Gehrels Swift observatory. We present optical/UV observations taken with the Swift Ultra-violet Optical Telescope, and compare them with the X-ray observations obtained with the Swift X-ray Telescope. We find that dust extinction affecting the optical/UV does not correlate with absorption due to neutral hydrogen that affects the X-ray emission. We suggest there is a small inhomogeneous high-density absorber containing a negligible amount of dust, close to the black hole. Overall, temporal variations in the optical/UV appear to trace those in the X-rays. During some epochs we observe an optical time-lag of (15-35) s. For both the optical/UV and X-rays, the amplitude of the variations correlates with flux, but this correlation is less significant in the optical/UV. The variability in the light curves may be produced by a complex combination of processes. Some of the X-ray variability may be due to the presence of a local, inhomogeneous and dust-free absorber, while variability visible in both the X-ray and optical/UV may instead be driven by the accretion flow: the X-rays are produced in the inner accretion disc, some of which are reprocessed to the optical/UV; and/or the X-ray and optical/UV emission is produced within the jet.