Explicit uniform bounds for Brauer groups of singular K3 surfaces

Let $k$ be a number field. We give an explicit bound, depending only on $[k:\mathbf{Q}]$, on the size of the Brauer group of a K3 surface $X/k$ that is geometrically isomorphic to the Kummer surface attached to a product of CM elliptic curves. As an application, we show that the Brauer-Manin set for such a variety is effectively computable. In addition, we prove an effective version of the strong Shafarevich conjecture for singular K3 surfaces by giving an explicit bound, depending only on $[k:\mathbf{Q}]$, on the number of $\mathbf{C}$-isomorphism classes of singular K3 surfaces defined over $k$.