Neutrinos act as probes of hadronic processes and offer a distinctive view into their astrophysical origins at high energies. When reaching energies on the PeV scale, $\nu_\tau$ interactions within the Earth can produce a significant flux of $\tau$-leptons. These $\tau$-leptons subsequently decay, generating upward-moving extensive air showers (EAS). Using the Earth as a target for neutrinos and the atmosphere as a signal generator effectively creates a detector with a mass $\gg$ gigaton. $\nu$SpaceSim is a comprehensive simulation developed to model all the relevant physical processes that describe the neutrino-induced, Earth-emergent lepton chain. The simulation models neutrino interactions inside the Earth that produce leptons, the propagation of the leptons through the Earth into the atmosphere, and their decay, forming composite EAS. Next, it models the generation of air optical Cherenkov and radio signals from these showers, including the propagation and attenuation of these signals through the atmosphere, accounting for effects such as clouds and the ionosphere. Finally, the simulation models the detector response according to the parameters defined by the user (such as altitude, effective area, frequency band...). Through this end-to-end simulation, $\nu$SpaceSim aims to help design the next generation of balloon- and space-based experiments, to estimate the exposure of ground-based experiments to these showers, and to understand the data from recent experiments such as EUSO-SPB2 and ANITA.