We present optical and near-infrared colour maps of the central regions of bulges of S0 and spiral galaxies obtained with WFPC2 and NICMOS on the Hubble Space Telescope (HST). By combined use of HST and ground-based data, the colour information spans a region from a few tens of pc to a few kpc. In almost all galaxies, the colour profiles in the central 100-200pc become more rapidly redder. We attribute the high central colour indices to a central concentration of dust. We infer an average extinction at the centre of AV=0.6-1.0mag. Several objects show central dust rings or discs at subkpc scales similar to those found by others in giant ellipticals. For galactic bulges of types S0 to Sb, the tightness of the B-I versus I-H relation suggests that the age spread among bulges of early-type galaxies is small, at most 2Gyr. Colours at 1Reff, where we expect extinction to be negligible, are similar to those of elliptical galaxies in the Coma cluster, suggesting that these bulges formed at the same time as the bright galaxies in Coma. Furthermore, the galaxy ages are found to be independent of their environment. As it is likely that Coma was formed at redshift z>3, our bulges, which are in groups and in the field, must also have been formed at this epoch. Bulges of early-type spirals cannot be formed by secular evolution of bars at recent epochs, because such bulges would be much younger. There are three galaxies of type Sbc and later; their bulges are younger and could perhaps arise from secular evolution of transient bars. Our results are in good agreement with semi-analytic predictions, which also predict that bulges, in clusters and in the field, are as old as giant ellipticals in clusters.