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cd-sIn 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 with more isolated dwarf 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 satellite dwarfs after infall.  We combine the observed 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 determine the typical timescales over which environmental processes quench satellite dwarf galaxies after infall. galaxies.  The quenching timescales at $\mstar<10^8\msun$ are short: short,  $< 2 - 3 \gyr$, comparable to a virial crossing time.  This timescale depends depending  somewhat on whether environmental preprocessing in lower-mass groupsprior to MW/M31 infall  is important. important, and are comparable to a virial-radius crossing time.  We compare with timescales for more massive satellites from the literature, which suggests implies  that environmental quenching timescale increases rapidly with satellite mass to $\approx 9.5\gyr$ $\approx9.5\gyr$  at $\mstar \approx 10 ^ 9 \msun$ $\mstar\approx 10^9\msun$,  but it  then rapidly decreases with mass to $<5\gyr$ at $\mstar>5\times 10^9\msun$. Thus, satellites with masses similar to the Magellanic Clouds exhibit the longest environmental quenching timescales.