We study the energy dependence of the total and diffractive neutrino-nucleon and neutrino-nucleus cross sections at very high energies. The calculation employs the QCD dipole model and the small-$x$ nonlinear Balitsky-Kovchegov evolution. We show the sensitivity of the nuclear effect quantification on the nuclear setup, and predict up to $\sim 10\%$ nuclear suppression in the inclusive neutrino-oxygen scattering stemming from the nonlinear evolution. Diffractive contribution to the total scattering is small, which is only few percentage. The $\left|q\bar{q}g\right>$ componnent of the $W^{\pm}$ boson is found to contribute significantly to the diffractive process, which reaches up to $\sim 40\%$ of the diffractive cross section.