Let$Y$be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of$\mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes$\text{Br}\,Y/\text{Br}_{1}\,Y$is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Néron–Severi lattices. Over a field of characteristic$0$, we prove that the existence of a strong uniform bound on the size of the odd torsion of$\text{Br}Y/\text{Br}_{1}Y$is equivalent to the existence of a strong uniform bound on integers$n$for which there exist non-CM elliptic curves with abelian$n$-division fields. Using the same methods we show that, for a fixed prime$\ell$, a number field$k$of fixed degree$r$, and a fixed discriminant of the geometric Néron–Severi lattice,$\#(\text{Br}Y/\text{Br}_{1}Y)[\ell ^{\infty }]$is bounded by a constant that depends only on$\ell$,$r$, and the discriminant.