Lags measured from correlated X-ray/UV/optical monitoring of AGN allow us to determine whether UV/optical variability is driven by reprocessing of X-rays or X-ray variability is driven by UV/optical seed photon variations. We present the results of the largest study to date of the relationship between the X-ray, UV and optical variability in an AGN with 554 observations, over a 750 d period, of the Seyfert 1 galaxy NGC 5548 with Swift. There is a good overall correlation between the X-ray and UV/optical bands, particularly on short time-scales (tens of days). The UV/optical bands lag the X-ray band with lags which are proportional to wavelength raised to the power 1.23 +/- 0.31. This power is very close to the power (4/3) expected if short time-scale UV/optical variability is driven by reprocessing of X-rays by a surrounding accretion disc. The observed lags, however, are longer than expected from a standard Shakura-Sunyaev accretion disc with X-ray heating, given the currently accepted black hole mass and accretion rate values, but can be explained with a slightly larger mass and accretion rate, and a generally hotter disc. Some long-term UV/optical variations are not paralleled exactly in the X-rays, suggesting an additional component to the UV/optical variability arising perhaps from accretion rate perturbations propagating inwards through the disc.