Uniform bounds on S-integral torsion points for Gm and elliptic curves

Let K be a number field, S a finite set of places. For Gm 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 Gm, 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 <= D.