We report the results of intensive X-ray, UV and optical monitoring of the Seyfert 1 galaxy NGC 4593 with Swift. There is no intrinsic flux-related spectral change in the the variable components in any band with small apparent variations due only to contamination by a second constant component, possibly a (hard) reflection component in the X-rays and the (red) host galaxy in the UV/optical bands. Relative to the shortest wavelength band, UVW2, the lags of the other UV and optical bands are mostly in agreement with the predictions of reprocessing of high energy emission from an accretion disc. The U-band lag is, however, far larger than expected, almost certainly because of reprocessed Balmer continuum emission from the more distant broad line region gas. The UVW2 band is well correlated with the X-rays but lags by ~6x more than expected if the UVW2 results from reprocessing of X-rays on the accretion disc. However, if the lightcurves are filtered to remove variations on timescales >5d, the lag approaches the expectation from disc reprocessing. MEMEcho analysis shows that direct X-rays can be the driver of most of the variations in the UV/optical bands as long as the response functions for those bands all have long tails (up to 10d) in addition to a strong peak (from disc reprocessing) at short lag (