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 H{alpha} 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 >5{sigma} 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.