This is a old question but I think it is still worthwile to supply a reference to a much more general statement (taken from subsection 2.3.3 of "The Theory of Infinite Soluble Groups" by John C. Lennox and Derek J. S. Robinson).

Let $\theta(x_1,\dots, x_n)$ be a word in the variables $x_1,\dots, x_n$.
If $H_1,\dots H_n$ are subgroups of a group $G$, let $\theta(H_1,\dots, H_n)$ be the subgroup generated by all $\theta(h_1,\dots, h_n)$
where $h_i\in H_i$. Then a result of Philip Hall says:

** Theorem** Let $G$ be a finitely generated nilpotent group with subgroups $H_i\le K_i$,
$i=1,\dots, n$ such that each index $|H_i:K_i|=m_i$ is finite. Then for every $n$-variable word $\theta$
the index $|\theta(H_1,\dots, H_n):\theta(K_1,\dots, K_n)|$ is finite and
it divides some power of $m_1 m_2 · · · m_n$.

In our case $n=1$ and the word $\theta$ is the commutator. Thus if $|G:H|=m$, then $|G^\prime: H^\prime|=m^r$ for some nonnegative interger $r$ (which could be $0$ as happens e.g. when $G$ is abelian).