There has been a recent interest in superconductor-magnetic insulator hybrid Rashba nanowire setups for potentially hosting Majorana zero modes at smaller external Zeeman fields. Using the Keldysh non-equilibrium Green's function technique, we develop a detailed quantum transport approach that accounts for the complex interplay between the quasiparticle dynamics in the superconductor-magnetic insulator bilayer structure, and the transport processes through the semiconducting Rashba nanowire. We provide a detailed analysis of three terminal setups to probe the local and non-local conductance spectra in both the pristine as well as the disordered nanowire setups. We uncover the Majorana conductance quantization scaling with the bilayer coupling and the signatures of the gap closing and reopening followed by the emergence of near-zero energy states, which can be attributed to topological zero modes in the clean limit. In the presence of a smoothly varying disorder potential, trivial Andreev bound states may form with signatures reminiscent of topological zero modes in the form of a premature gap closure in the conductance spectra. Our results therefore provide transport-based analysis of the operating regimes that support the formation of Majorana modes in these hybrid systems of current interest, while investigating the effect of disorder on experimentally relevant device structures.