  Let $L$ be a number field and let $E/L$ be an elliptic curve with complex
multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary
quadratic field $K$. We use class field theory and results of Skorobogatov and
Zarhin to compute the transcendental part of the Brauer group of the abelian
surface $E\times E$. The results for the odd order torsion also apply to the
Brauer group of the K3 surface $\textrm{Kum}(E\times E)$. We describe
explicitly the elliptic curves $E/\mathbb{Q}$ with complex multiplication by
$\mathcal{O}_K$ such that the Brauer group of $E\times E$ contains a
transcendental element of odd order. We show that such an element gives rise to
a Brauer-Manin obstruction to weak approximation on $\textrm{Kum}(E\times E)$,
while there is no obstruction coming from the algebraic part of the Brauer
group.
