  Given a number field $F_0$ that contains no Hilbert class field of any
imaginary quadratic field, we show that under GRH there exists an effectively
computable constant $B:=B(F_0)\in\mathbb{Z}^+$ for which the following holds:
for any finite extension $L/F_0$ whose degree $[L:F_0]$ is coprime to $B$, one
has for all elliptic curves $E_{/F_0}$ that the $L$-rational torsion subgroup
$E(L)[\textrm{tors}]=E(F_0)[\textrm{tors}]$. This generalizes a previous result
of Gonz\'{a}lez-Jim\'{e}nez and Najman over $F_0=\mathbb{Q}$.
  Towards showing this, we also prove a result on relative uniform divisibility
of the index of a mod-$\ell$ Galois representation of an elliptic curve over
$F_0$. Additionally, we show that the main result's conclusion fails when we
allow $F_0$ to have rationally defined CM, due to the existence of
$F_0$-rational isogenies of arbitrarily large prime degrees satisfying certain
congruency conditions.
