We present and analyse optical and ultraviolet (UV) colours for passive and optically-red Coma cluster galaxies for which we have spectroscopic age and element abundance estimates. Our sample of 150 objects covers a wide range in mass, from giant ellipticals down to the bright end of the dwarf-galaxy regime. Galaxies with ongoing star formation have been removed using strict Hα emission-line criteria. We focus on the colours FUV−i, NUV−i, FUV−NUV, u*−g and g−i. We find that all of these colours are correlated with both luminosity and velocity dispersion at the >5σ level, with FUV−i and FUV−NUV becoming bluer with increasing ‘mass’ while the other colours become redder. We perform a purely empirical analysis to assess what fraction of the variation in each colour can be accounted for by variations in the average stellar populations, as traced by the optical spectra. For the optical colours, u*−g and g−i, most of the observed scatter (∼80 per cent after allowing for measurement errors and for systematic errors in u*−g) is attributable to stellar population variations, with colours becoming redder with increasing age and metallicity (Mg/H). The FUV−i colour becomes bluer with increasing age and with increasing Mg/H, favouring the ‘metal-rich single-star’ origin for the UV upturn. However, correlations with the optically-dominant stellar populations account for only about half of the large observed scatter. We propose that the excess scatter in FUV−i may be due to a varying proportion of ancient stars in galaxies with younger [simple stellar population (SSP) equivalent] average ages. The NUV−i colour is sensitive to SSP-equivalent age and Mg/H (in the same sense as optical colours), but also exhibits excess scatter that can be attributed to ‘leakage’ of the far-UV-dominant (FUV-dominant) old hot population. After applying a correction based on the FUV−i colour, the much of the remaining variance in NUV−i is attributable to variations in the spectroscopic parameters, similar to the results for optical colours. Finally, the FUV−NUV colour is surprisingly well behaved, showing strong correlations with age and metallicity, and little residual scatter. Interpreting this colour is complicated, however, since it mixes the effects of the main-sequence turn-off, in the near-UV, with the variation in the hot post-red giant branch content dominating the FUV.