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In the Local Group, nearly all of the dwarf galaxies ($\mstar < 10 ^ 9 \msun$) ($\mstar<10^9\msun$)  that are satellites within $300 \kpc$ $300\kpc$  (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 the more isolated dwarf galaxies at larger distances, 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 after infall.  We combine the observed high  quiescent fractions of 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 inferstatistically  the typical  timescales over which environmental processes quench  satellite dwarf galaxies are environmentally quenched. after infall.  The quenching timescales at $\mstar<10^8\msun$ are short: $< 2 - 3 \gyr$, depending on whether environmental preprocessing in lower-mass groups is important.  We compare with the timescales for more massive satellites from the literature, which suggests that environmental quenching timescales are longest ($\approx 9.5\gyr$) for satellites increase rapidly with satellite mass to $\approx 9.5\gyr$  at $\mstar \approx 10 ^ 9 \msun$ and but then rapidly  decrease to $<5\gyr$ $\less 5\gyr$  at $\mstar>5\times 10^9\msun$. Thus, Magellanic-Cloud-mass satellites have the longest environmental quenching timescales.