We combine classical and ring polymer molecular dynamics simulations with the molecular jump model to provide a molecular description of the nuclear quantum effects (NQEs) on water reorientation and hydrogen-bond dynamics in liquid H2O and D2O. We show that while the net NQE is negligible in D2O, it leads to a similar to 13% acceleration in H2O dynamics compared to a classical description. Large angular jumps exchanging hydrogen-bond partners are the dominant reorientation pathway (just as in a classical description); the faster reorientation dynamics arise from the increased jump rate constant. NQEs do not change the jump amplitude distribution, and no significant tunneling is found. The faster jump dynamics are quantitatively related to decreased structuring of the 00 radial distribution function when NQEs are included. This is explained, via a jump model analysis, by competition between the effects of water's librational and OH stretch mode zero-point energies on the hydrogen-bond strength.