The unexpected rising flux of early-type galaxies at decreasing ultraviolet (UV) wavelengths is a long-standing mystery. One important observational constraint is the correlation between UV-optical colours and Mg2 line strengths found by Burstein et al. The simplest interpretation of this phenomenon is that the UV strength is related to the Mg line strength. Under this assumption, we expect galaxies with larger Mg gradients to have larger UV colour gradients. By combining UV imaging from GALEX, optical imaging from MDM and SAURON integral-field spectroscopy, we investigate the spatially resolved relationships between UV colours and stellar population properties of 34 early-type galaxies from the SAURON survey sample. We find that galaxies with old stellar populations show tight correlations between the far-UV (FUV) colours (FUV -V and FUV - NUV) and the Mg b index, Hβ index and metallicity [Z/H]. The equivalent correlations for the Fe5015 index, α-enhancement [α/Fe] and age are present but weaker. We have also derived logarithmic internal radial colour, measured line strength and derived stellar population gradients for each galaxy and again found a strong dependence of the FUV -V and FUV - NUV colour gradients on both the Mg b line strength and the metallicity gradients for galaxies with old stellar populations. In particular, global gradients of Mg b and [Z/H] with respect to the UV colour [e.g. Δ(Mg b)/Δ(FUV - NUV) and Δ[Z/H]/Δ(FUV - NUV)] across galaxies are consistent with their local gradients within galaxies, suggesting that the global correlations also hold locally. From a simple model based on multiband colour fits of UV upturn and UV-weak galaxies, we have identified a plausible range of parameters that reproduces the observed radial colour profiles. In these models, the centres of elliptical galaxies, where the UV flux is strong, are enhanced in metals by roughly 60 per cent compared to UV-weak regions.