With the upcoming Run 3 of the LHC, the FASERν and SND@LHC detectors will start a new era of neutrino physics using the far-forward high-energy neutrino beam produced in collisions at ATLAS. This emerging LHC neutrino physics program requires reliable estimates of the LHC's forward neutrino fluxes and their uncertainties. In this paper we provide a new fast-neutrino flux simulation, implemented as a RIVET module, to address this issue. We present the expected energy distributions going through the FASERν and SND@LHC detectors based on various commonly used event generators, analyze the origin of those neutrinos, and present the expected neutrino event rates.