Studies of the structure of hydroxides under pressure using neutron diffraction reveal that the high concentration of hydrogen is distributed in a disordered network. The disorder in the hydrogen-bond network and possible phase transitions are reported to occur at pressures within the range accessible to experiments for layered calcium hydroxides, which are considered to be exemplary prototype materials. In this study, the static and dynamical properties of these layered hydroxides are investigated using a quantum approach describing nuclear motion, shown herein to be required particularly when studying diffusion processes involving light hydrogen atoms. The effect of high-pressure on the disordered hydrogen-bond network shows that the protons tunnel back and forth across the barriers between three potential minima around the oxygen atoms. At higher pressures the structure has quasi two-dimensional layers of hydrogen atoms, such that at low temperatures this causes the barrier crossing of the hydrogen to be significantly rarefied. Furthermore, for moderate values of both temperature and pressure this process occurs less often than the usual mechanism of proton transport via vacancies, limiting global proton diffusion within layers at high pressure.