Polynomial bounds on torsion from a fixed geometric isogeny class of elliptic curves

We show there exist polynomial bounds on torsion of elliptic curves which come from a fixed geometric isogeny class. More precisely, for an elliptic curve $E_0$ defined over a number field $F_0$, for each $\epsilon >0$ there exist constants $c_{\epsilon }:=c_{\epsilon }(E_0,F_0),C_{\epsilon }:=C_{\epsilon }(E_0,F_0)>0$ such that for any elliptic curve $E_{\mathrm{/}F}$ geometrically isogenous to $E_0$, if $E(F)$ has a point of order $N$ then $$N\le c_{\epsilon }\cdot [F:\mathbb{Q}{]}^{1\mathrm{/}2+\epsilon },$$ and one also has $$\mathrm{\#}E(F)[\text{tors}]\le C_{\epsilon }\cdot [F:\mathbb{Q}{]}^{1+\epsilon }\mathrm{.}$$