The LHC generates an intense beam of high-energy neutrinos in the forward direction, whose
scientific potential has been left unexploited for many years. The FASER and SND@LHC experiments, operating since 2023, have recently measured LHC neutrinos for the first time. In this contribution we discuss how to produce accurate predictions, including NLO QCD corrections and modern parton shower algorithms, for present and future LHC neutrino experiments, including those at the proposed Forward Physics Facility (FPF). To this end, the energy and rapidity distribution of the LHC neutrinos is encoded in a LHAPDF grid interfaced to the neutrino DIS event generator in the POWHEG-BOX-RES framework. This Monte Carlo tool enables the modelling of differential distributions that are sensitive to hadronic final states, initial- and final-state radiation, and realistic acceptance and selection cuts. As a first application, we deploy this event generator to compute fast-interpolation grids and carry out a first determination of the LHC forward neutrino fluxes directly from FASER data using the NNPDF fitting methodology.