  In the Local Group, nearly all of the dwarf galaxies (M_star < 10^9 M_sun)
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
rapid transformation of satellite dwarf galaxies after falling into the halos
of the MW or M31. We combine the observed quiescent fractions for satellites of
the MW and M31 with the infall times of satellites from the Exploring the Local
Volume in Simulations (ELVIS) suite of cosmological zoom-in simulations to
determine the typical timescales over which environmental processes within the
MW/M31 halos remove gas and quench star formation in low-mass satellite
galaxies. The quenching timescales for satellites with M_star < 10^8 M_sun are
short, < 2 Gyr, and quenching is more rapid at lower M_star. These satellite
quenching timescales can be 1 - 2 Gyr longer if one includes the time that
satellites were environmentally preprocessed by low-mass groups prior to MW/M31
infall. We compare with quenching timescales for more massive satellites from
previous works to synthesize the nature of satellite galaxy quenching across
the observable range of M_star = 10^{3-11} M_sun. The satellite quenching
timescale increases rapidly with satellite M_star, peaking at ~9.5 Gyr for
M_star ~ 10^9 M_sun, and the timescale rapidly decreases at higher M_star to <
5 Gyr at M_star > 5 x 10^9 M_sun. Overall, galaxies with M_star ~ 10^9 M_sun,
similar to the Magellanic Clouds, exhibit the longest quenching timescales,
regardless of environmental or internal mechanisms.
