The cosmic neutrino background (C$\nu$B) can be boosted to high energies due to scatterings with energetic cosmic rays (CRs) across cosmological scales. Previous calculations focused on neutral current incoherent and coherent elastic scatterings of cosmic-ray protons off relic neutrinos. However, charged current interactions and deep inelastic scatterings are also expected to occur, which enhance the boosted relic neutrino fluxes on Earth. Here, we compute the diffuse boosted cosmic neutrino background (DBC$\nu$B) arising from CRs at all redshifts in the Universe, accounting for neutral- and charged-current quasi-elastic scatterings and neutral-current deep inelastic scatterings. We find that a joint likelihood analysis of current neutrino telescopes already places an upper limit on the cosmic neutrino background overdensity on cosmological scales of $𝒪({10}^2-{10}^3)$, for a lightest neutrino mass of $m_{\nu }\gtrsim 0.1$ eV. We further show that a joint analysis of the proposed ultra-high-energy neutrino telescopes could test C$\nu$B overdensities of $𝒪(10)$, below the maximal value allowed by the energy budget of the Universe for $m_{\nu }\gtrsim 0.1$ eV, and approaching the $\Lambda$CDM expectation for the largest neutrino masses compatible with the KATRIN bound.