  Let $K$ be a number field, $S$ a finite set of places. For $\mathbb{G}_m$ or
an elliptic curve $E$ defined over $K$, we establish uniformity results on the
number of $S$-integral torsion points relative to a non-torsion point $\beta$,
as $\beta$ varies over number fields of bounded degree. In particular for
$\mathbb{G}_m$, if $D$ is a positive integer, we prove a uniform bound on the
degree of a torsion point $\zeta$ that is $S$-integral relative to a
non-torsion point $\beta$ with degree $\leq D$.
