High-fidelity simulation of multiscale multiphysics systems gives rise to large nonlinear algebraic systems whose solution by Newton-type methods constitutes a significant computational challenge. Classical homogenization alleviates this burden when a clear separation of scales is present; however, many practical applications exhibit only partial scale separation and therefore require strongly coupled multiscale solvers, which are often limited to periodic microstructures. We propose a Multi-Grid Two-Scale (MGTS) framework that balances accuracy and efficiency through a hierarchical formulation. The MGTS model combines a nonlinear coarse-scale upscaled problem with a linear fine-scale corrector obtained by linearizing the full system about the coarse-scale solution. This approach captures the dominant coarse- and fine-scale effects while confining the nonlinear iterations to the coarse grid. Moreover, the global solution of the fine-scale corrector eliminates the need for artificial interface conditions on local microscale subproblems. The resulting weak coupling between scales further enables efficient parallel-in-time implementations. The framework is applied to a nonlinear biphasic model of saturated poroelasticity in thin domains. The resulting MGTS formulation consists of a reduced-dimension homogenized model, derived via formal asymptotic analysis, that describes the effective nonlinear longitudinal behaviour, together with a linearized corrector that resolves transverse small-scale processes. The reliability of the MGTS approach is demonstrated numerically, and the influence of key parameters, including aspect ratio, boundary periodicity, and material heterogeneity, is systematically analysed to identify the regimes in which the MGTS method achieves optimal performance.