We present the most comprehensive two-dimensional numerical model to date for a nanoconfined bipolar electrochemical system. By accounting for the compact Stern layer and resolving the diffuse part of the electrical double layer at the bipolar electrode (BPE) surface and channel walls, our model captures the impact of surface polarization and ionic charge-screening effects on the heterogeneous charge-transfer kinetics, as well as nonlinear electrokinetic transport phenomena such as induced-charge electroosmosis and concentration polarization. We employ the Poisson-Nernst-Planck and Stokes flow system of equations, unified with generalized Frumkin-Butler-Volmer reaction kinetics, to describe water electrolysis reactions and the resulting transport of ions and dissolved gases in the confined BPE system. Our results demonstrate that under a sufficiently large applied electric field, the rapid reaction kinetics on our Pt BPE dynamically transition from charge-transfer-limited to mass-transfer-limited temporal regimes as regions depleted of redox species form and propagate outward from the respective BPE poles. This phenomenon was visualized experimentally with a pH-sensitive fluorescein dye and showed excellent phenomenological agreement with our numerical calculations, providing a foundation for further understanding and developing bipolar electrochemical processes in confined geometries. We introduce two prospective applications arising from our work: (1) a hybrid hydrodynamic/electrochemical peristaltic pump and (2) deducing information about chemical kinetics through simulation.