Using a sample of 4 galaxy clusters at $1.35 < z < 1.65$ and 10 galaxy clusters at $0.85 < z < 1.35$, we measure the environmental quenching timescale, $t_Q$, corresponding to the time required after a galaxy is accreted by a cluster for it to fully cease star formation. Cluster members are selected by a photometric-redshift criterion, and categorized as star-forming, quiescent, or intermediate according to their dust-corrected rest-frame colors and magnitudes. We employ a delayed-then-rapid quenching model that relates a simulated cluster mass accretion rate to the observed numbers of each type of galaxy in the cluster to constrain $t_Q$. For galaxies of mass $M_* gtrsim 10^{10.5}~ mathrm{M}_odot$, we find a quenching timescale of $t_Q=$ 1.24 Gyr in the $zsim1.5$ cluster sample, and $t_Q=$ 1.50 Gyr at $zsim1$. Using values drawn from the literature, we compare the redshift evolution of $t_Q$ to timescales predicted for different physical quenching mechanisms. We find $t_Q$ to depend on host halo mass such that quenching occurs over faster timescales in clusters relative to groups, suggesting that properties of the host halo are responsible for quenching high-mass galaxies. Between $z=0$ and $z=1.5$, we find that $t_Q$ evolves faster than the molecular gas depletion timescale and slower than an SFR-outflow timescale, but is consistent with the evolution of the dynamical time. This suggests that environmental quenching in these galaxies is driven by the motion of satellites relative to the cluster environment, although due to uncertainties in the atomic gas budget at high redshift, we cannot rule out quenching due to simple gas depletion.
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