Explicit uniform bounds for Brauer groups of singular K3 surfaces

Let $k$ be a number field. We give an explicit bound, depending only on $[k:mathbf{Q}]$ and the discriminant of the N{e}ron--Severi lattice, 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 isogenous CM elliptic curves. As an application, we show that the Brauer--Manin set for such a variety is effectively computable. Conditional on the Generalised Riemann Hypothesis, we also 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. In addition, we show how to obtain a bound, depending only on $[k:mathbf{Q}]$, on the number of $mathbf{C}$-isomorphism classes of singular K3 surfaces defined over $k$, thus proving an effective version of the strong Shafarevich conjecture for singular K3 surfaces.