The phase transition from ice VII to ice X under extreme pressures is an example where quantum proton delocalization coexists with classical thermal fluctuations. We investigate this transition, including quantum effects on the nuclear motion through adapted Langevin dynamics. This approach, which allows us to follow the semiclassical trajectories of protons, provides excellent agreement with experimental vibrational spectra indicating a transition pressure of about 65 GPa. Furthermore, we map the full dynamical problem onto a pressure-dependent, one-dimensional mean-field potential for the proton. By solving exactly the corresponding Schrödinger equation, we disentangle tunneling and quantum delocalization from classical thermal effects and identify the transition through the topological changes of the proton ground state and its susceptibility. The process is dominated by quantum effects even at ambient temperature and can be considered to be a paradigmatic case of a quantum-driven phase transition.