This is the second of a series of three papers exploring the connection between the multiwavelength properties of AGN in nearby early-type galaxies and the characteristics of their hosts. In Capetti et al. (2005) we presented a study of the surface brightness profiles for the 65 objects with available archival HST images out of the 116 radio-detected galaxies. We classified early-type galaxies into ``core and ``power-law galaxies, discriminating on the basis of the slope of their nuclear brightness profiles. Here we focus on the 29 core galaxies (hereafter CoreG). We used HST and Chandra data to isolate their nuclear emission. The CoreG invariably host radio-loud nuclei, with an average radio-loudness parameter of Log R = 3.6. The optical and X-ray nuclear luminosities correlate with the radio-core power, smoothly extending the analogous correlations already found for low luminosity radio-galaxies. This supports the interpretation of a common non-thermal origin of the nuclear emission also for CoreG. The luminosities of the nuclear sources, most likely dominated by jet emission, set firm upper limits, as low as L/L_Edd = 10^{-9} in both the optical and X-ray band, on any emission from the accretion process. The similarity of CoreG and LLRG indicates that they are drawn from the same population of early-type galaxies. LLRG represent only the tip of the iceberg associated with (relatively) high activity levels, with CoreG forming the bulk of the population. A minimum black hole mass of M_{BH} = 10^8 M_sun is apparently associated with the radio-loud nuclei in both CoreG and LLRG, but this effect must be tested on a sample of less luminous galaxies, likely to host smaller black holes. ABRIDGED.
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