It's true for finitely generated nilpotent groups. Let $G$ be such a group with lower central series $G=G_1 > G_2 > \cdots > G_{c+1}=1$, and let $H<G$ with $|G:H|$ finite. . By induction on the class $c$, we can assume that $|G/G_c: H'G_c|$ is finite, so it suffices to prove that $|G_c:H' \cap G_c|$ is finite.

The commutator map induces a surjective homomorphism $G/G' \otimes G_{c-1}/G_c \to G_c$. Since $|G/G':G'H'/G'|$ and $|G_{c-1}/G_c:(H' \cap G_{c-1})G_c/G_c|$ are both finite, the image of the restriction of this map to $G'H'/G' \otimes (H' \cap G_{c-1})G_c/G_c$, which is equal to $H' \cap G_c$, has finite index in $G_c$, and we are done.

It is not rue in general for infinitely generated nilpotent groups. For example, let $G$ be a covering group of an infinitely generated elementary abelian $p$-group, and let $H$ be a subgroup of $G$ of index $p$ that contains $G'$.