The emergence of two-dimensional topological materials, particularly the group-14 monolayers known as silicene, germanene, and stanene has opened promising pathways for next-generation nanoelectronics and spintronics. Their buckled honeycomb structure and strong spin-orbit coupling allow for bandgap engineering via a perpendicular electric field, leading to topological phase transitions (TPTs) from non-trivial to trivial insulating states. However, precise determination of the critical electric field E_zᶜʳ at which these transitions occur remains challenging, with tight-binding models often underestimating these values. Here, we present a first-principles framework that combines density-functional theory (DFT), maximally localized Wannier functions, and evolution of the Wannier charge centers (WCC) to accurately characterize TPTs in silicene, germanene, and stanene through the Z₂ topological invariant. In contrast to earlier work, at each electric-field strength we run fully self-consistent ab initio simulations to obtain the screened electronic structure, accounting for the material's dielectric response from both electrons and ions. From these converged results we construct a Wannier tight-binding Hamiltonian at each electric field strength, which then enables a gauge-invariant calculation of the Z₂ topological invariant. This methodology yields significantly more accurate numerical predictions of E_zᶜʳ, 0.020 and 0.250 V/Å for silicene and germanene, respectively. Compared to previous approaches, our framework delivers a marked quantitative improvement for predicting topological phase boundaries, essential for guiding the design of topological field-effect transistors and electrostatically controlled quantum devices based on two-dimensional materials.