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In the Local Group, nearly all of the dwarf galaxies ($\mstar<10^9\msun$) that are satellites within $300\kpc$ (the virial radius) of the Milky Way (MW) and Andromeda (M31) have quiescent star formation and little-to-no cold gas.  This contrasts strongly withthe  more isolated dwarf galaxies at larger distances, galaxies,  which are almost all actively star-forming and gas-rich. This near dichotomy implies that environmental processes within the halos of the MW and M31 \emph{rapidly} remove gas and quench star formation in satellites satellite dwarfs  after infall. We combine the observedhigh  quiescent fractions for satellites of the MW/M31 with the virial-infall times of satellites in the ELVIS suite of cosmological simulations of MW/M31-like halos to infer determine  the typical timescales over which environmental processes quench satellite dwarf galaxies after infall. The quenching timescales at $\mstar<10^8\msun$ are short: $< 2 - 3 \gyr$, depending comparable to a virial crossing time.  This timescale depends somewhat  on whether environmental preprocessing in lower-mass groups prior to MW/M31 infall  is important. We compare withthe  timescales for more massive satellites from the literature, which suggests that environmental quenching timescales increase timescale increases  rapidly with satellite mass to $\approx 9.5\gyr$ at $\mstar \approx 10 ^ 9 \msun$ but then rapidly decrease decreases with mass  to $<5\gyr$ at $\mstar>5\times 10^9\msun$. Thus,Magellanic-Cloud-mass  satellites have with masses similar to the Magellanic Clouds exhibit  the longest environmental quenching timescales.