Surface abundance observations of halo stars hint at the occurrence of r-process nucleosynthesis at low metallicity ([Fe/H] < -3), possibly within the first 108 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 108 yr even without a third object is already high. Our results provide additional support to the compact-binary merger scenario for r-process nucleosynthesis.