We present a complete analysis of the Fundamental Plane (FP) of early-type galaxies (ETGs) in the nearby Universe (z < 0.1). The sample, as defined in Paper I, comprises 39993 ETGs located in environments covering the entire domain in local density (from field to cluster). We derive the FP in the grizYJHK wavebands with a detailed discussion on fitting procedure, bias due to selection effects and bias due to correlated errors on the effective parameters, re and <μ>e, as key factors in obtaining meaningful FP coefficients. Studying the Kormendy relation (KR) we find that its slope varies from g (3.44 +/- 0.04) through K (3.80 +/- 0.02) implying that smaller size ETGs have a larger ratio of optical to near-infrared (NIR) radii than galaxies with larger re. We also examine the Faber-Jackson (FJ) relation and find that its slope is similar for all wavebands, within the uncertainties, with a mean value of 0.198 +/- 0.007. Writing the FP equation as logre = alogσ0 + b<μ>e + c, we find that the `a' varies from 1.38 +/- 0.02 in g to 1.55 +/- 0.02 in K, implying a 12 per cent variation across the grizYJHK wavelength baseline. The corresponding variation of `b' is negligible (b ~ 0.316), while `c' varies by ~10 per cent. We show that the waveband dependence of the FJ and KR results from the complex variation of the distribution of galaxies in the face-on projection of the FP as well as by the change of FP coefficients with waveband. We find that `a' and `b' become smaller for higher Sersic index and larger axial ratios, independent of the waveband. This suggests that these variations are likely to be related to differences in structural and dynamical (rather than stellar population) properties of ETGs. It is noticeable that galaxies with bluer colours and disc-like isophotes have smaller `b', with the effect decreasing smoothly from g through K. Considering a power-law relation between mass-to-light ratio and (dynamical) mass, M/L ~ Mγ, we estimate γ from the FP coefficients in grizYJHK. The γ decreases from 0.224 +/- 0.008 in g to 0.186 +/- 0.009 in K band. Using the γ values, we estimate the variation of age and metallicity of the stellar populations present in massive galaxies per decade in stellar mass. This analysis shows that in the NIR the tilt of the FP is not due to stellar population variation, and that ETGs have coeval stellar populations with an age variation of a few per cent per decade in mass, and a corresponding metallicity increase of ~23 per cent. We also show that current semi-analytical models of galaxy formation reproduce very well these amounts of variation of age and metallicity with respect to stellar mass.