The field of twistronics, recently extended to twisted bilayer cuprate systems, has generated considerable interest due to the prediction of d+id' states close to a 45^∘ twist, as suggested by multiple experimental probes. Theoretical studies of these systems using a variety of models and many-body theoretical techniques have revealed multiple competing order parameters. The large variability in experimental results and theoretical predictions demands an accurate microscopic model for these systems, in particular a more realistic description of the interlayer tunneling. In this study, we fit the in-plane hopping and compute the interlayer tunneling amplitudes by calculating the orbital overlap between d_(x²-y²) wave functions at each site of the twisted square-lattice model. We observe sign-changing tunneling amplitudes, a strong twist-angle dependence of the direct tunneling term, and long-range tunneling amplitudes that often dominate over direct tunneling. We investigate the band structure, Fermi surfaces, and spectral functions in comparison with those obtained from the phenomenological model and find a substantial redistribution of spectral weight that modifies the competition between different superconducting ground states. We determine the phase diagram as a function of twist angle, filling, and direct tunneling strength, which is controlled by the c-axis distance between the monolayers, and observe a reduction in the region of time-reversal-symmetry-breaking (TRSB) d+id' states compared with the phenomenological model. Finally, we investigate the Josephson critical current across the phase diagram and find strongly suppressed critical currents close to a 45^∘ twist, consistent with experimental observations.