  We perform a detailed phenomenological study of high-energy neutrino deep
inelastic scattering (DIS) focused on LHC far-forward experiments such as
FASER$\nu$ and SND@LHC. To this aim, we parametrise the neutrino fluxes
reaching these LHC far-forward experiments in terms of `neutrino PDFs' encoding
their energy and rapidity dependence by means of the LHAPDF framework. We
integrate these neutrino PDFs in the recently developed POWHEG-BOX-RES
implementation of neutrino-induced DIS to produce predictions accurate at
next-to-leading order (NLO) in the QCD coupling matched to parton showers (PS)
with Pythia8. We present NLO+PS predictions for final-state distributions
within the acceptance of FASER$\nu$ and SND@LHC as well as for two experiments
of the proposed Forward Physics Facility (FPF), FASER$\nu$2 and FLArE. We
quantify the impact of NLO QCD corrections, of the parton showering and
hadronisation settings in Pythia8, of the QED shower, and of the incoming
neutrino flavour for the description of these observables, and compare our
predictions with the GENIE neutrino event generator. Our work demonstrates the
relevance of modern higher-order event generators to achieve the key scientific
targets of the LHC neutrino experiments.
