<p>The Migdal–Eliashberg (ME) formalism provides a reliable framework for describing phonon-mediated superconductivity in the adiabatic regime, where the Fermi energy exceeds the characteristic phonon energy. Here, we extend this framework by incorporating first-order vertex corrections to the electron–phonon (e–ph) interaction and evaluating their impact on H₃S and Pb using first-principles calculations. For H₃S, where the adiabatic assumption breaks down, vertex corrections to the e–ph coupling are substantial. Combined with phonon anharmonicity and the energy dependence of the electronic density of states, these effects yield a predicted critical temperature <i>T</i>_c in excellent agreement with experiment. In contrast, for elemental Pb, where the adiabatic limit holds, vertex corrections are negligible, and the calculated <i>T</i>_c and gap closely match standard ME predictions. This study demonstrates the importance of non-adiabatic corrections in strongly coupled high-<i>T</i>_c hydrides and establishes a robust first-principles framework for accurately predicting superconducting properties across different regimes.</p>