We measure the real part ε^{′} of the dielectric permittivity of beryl crystals with heavy water molecules D_{2}O confined in nanosized cages formed by an ionic crystal lattice. The experiments are performed at a frequency of 1 MHz in the temperature interval from 300 down to 4 K under different hydrostatic pressures up to P=6.3 GPa. At high temperatures, a Curie-Weiss-like increase of ε^{′}(T) is observed upon cooling. Application of pressure leads to flattening of ε^{′}(T) at low temperatures due to quantum effects, i.e., tunneling of deuterium atoms in the hexagonal localizing potential. Analyzing the temperature behavior of ε^{′} with the Barrett expression allows us to obtain pressure dependencies of the quantum temperature T_{1}, the Curie-Weiss temperature T_{C}, and the Barrett constant C. The increase of T_{1} observed up to 4 GPa is associated with an enhanced azimuthal tunneling of the confined water molecules through the barriers of the potential. For P>4 GPa, T_{1}(P) levels off since the barriers disappear. Any further pressure increase does not affect the tunneling rate because of the absence of a barrier. The behavior is modeled by solving the Schrödinger equation for the water molecule in the azimuthal potential numerically. Small negative values of T_{C}≈−10 K obtained for P<4 GPa indicate the antiferroelectric ordering tendency of the water dipoles localized in the crystalline nanochannels. For higher pressure, a strong decrease of T_{C} toward negative values is observed that would correspond to the enhanced interdipole coupling strength, which is however hard to explain in the present case, and thus calls for additional theoretical and experimental studies.