  Surface abundance observations of halo stars hint at the occurrence of
$r$-process nucleosynthesis at low metallicity ($\rm{[Fe/H]< -3}$), possibly
within the first $10^8$ yr after the formation of the first stars. Possible
loci of early-Universe $r$-process nucleosynthesis are the ejecta of either
black hole--neutron star or neutron star--neutron star binary mergers. Here we
study the effect of the inclination--eccentricity oscillations raised by a
tertiary (e.g. a star) on the coalescence time scale of the inner compact
object binaries. Our results are highly sensitive to the assumed initial
distribution of the inner binary semi-major axes. Distributions with mostly
wide compact object binaries are most affected by the third object, resulting
in a strong increase (by more than a factor of 2) in the fraction of fast
coalescences. If instead the distribution preferentially populates very close
compact binaries, general relativistic precession prevents the third body from
increasing the inner binary eccentricity to very high values. In this last
case, the fraction of coalescing binaries is increased much less by tertiaries,
but the fraction of binaries that would coalesce within $10^8$ yr even without
a third object is already high. Our results provide additional support to the
compact object merger scenario for $r$-process nucleosynthesis.
