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In the Local Group, nearly all of the dwarf galaxies ($\mstar\lesssim10^9\msun$) 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 comparatively isolated dwarf galaxies, which are almost all actively star-forming and gas-rich.  This near dichotomy implies a \emph{rapid} transformation after falling into the halo halos  of the MW/M31. MW or M31.  We combine the observed quiescent fractions for satellites of the MW/M31 MW and M31  with the infall times of satellites from the ELVIS suite of cosmological simulations to determine the typical timescales over which environmental processes remove gas and quench star formation in satellite dwarf galaxies. The quenching timescales at $\mstar<10^8\msun$ are short, $\lesssim2\gyr$, and decrease with decreasing satellite mass.  %and are comparable to a virial-radius crossing time.  These quenching  timescales can increase be  $1-2\gyr$ longer  if environmental preprocessing in lower-mass groups prior to MW/M31 infall is important. Comparing We compare  with timescales for more massive satellites from previous works works, which  indicates that the environmental quenching timescale increases rapidly with satellite mass and peak mass, peaking  at $\approx9.5\gyr$ for $\mstar\approx10^9\msun$, and  rapidly decreases at higher satellite mass to less than $5\gyr$ at $\mstar>5\times10^9\msun$. Thus, satellites with masses similar to the Magellanic Clouds exhibit the longest environmental quenching timescales.