Using HST NICMOS 2 observations we have measured 1.6 μm near-infrared nuclear luminosities of 100 3CR radio galaxies with z < 0.3, by modeling and subtracting the extended emission from the host galaxy. We performed a multiwavelength statistical analysis (including optical and radio data) of the properties of the nuclei following classification of the objects into FR I and FR II, and low-ionization galaxies (LIGs), high-ionization galaxies (HIGs), and broad-line objects (BLOs) using the radio morphology and optical spectra, respectively. The correlations among near-infrared, optical, and radio nuclear luminosity support the idea that the nearinfrared nuclear emission of FR Is has a non-thermal origin. Despite the difference in radio morphology, the multiwavelength properties of FR II LIG nuclei are statistically indistinguishable from those of FR Is, an indication of a common structure of the central engine. All BLOs show an unresolved near-infrared nucleus and a large near-infrared excess with respect to FR II LIGs and FR Is of equal radio core luminosity. This requires the presence of an additional (and dominant) component other than the non-thermal light. Considering the shape of their spectral energy distribution, we ascribe the origin of their near-infrared light to hot circumnuclear dust. A near-infrared excess is also found in HIGs, but their nuclei are substantially fainter than those of BLO. This result indicates that substantial obscuration along the line of sight to the nuclei is still present at 1.6 μm. Nonetheless, HIG nuclei cannot simply be explained in terms of dust obscuration: a significant contribution from light reflected in a circumnuclear scattering region is needed to account for their multiwavelength properties.