Since $X$ is free, the Bredon cohomology of $X\times Y$ agrees with the usual cohomology of the orbit space. There is a homotopy pullback square
$$
\begin{array}{ccc}
(X\times Y)/G & \rightarrow & Y_{hG} \\
\downarrow & & \downarrow \\
X/G & \rightarrow & BG
\end{array}
$$
where $Y_{hG}$ denotes the Borel construction, or homotopy orbits. The Eilenberg-Moore
spectral sequence of this homotopy pullback square is a Kunneth spectral sequence abutting to what you want, but it starts from the Bredon cohomology of $X$ (which is the same as the usual cohomology of $X/G$) and the cohomology of the Borel construction on $Y$, regarded as modules over $H^*(BG)$. As far as I understand, convergence is not guaranteed either...
So this is probably not what you want, but I thought it might be worth mentioning.