Let $X$ and $Y$ be CW complexes. Let $X\times Y$ be the usual product (in $\mathbf{Top}$, the category of all spaces), and let $X\times_k Y$ be the $k$-ification of $X\times Y$, that is the topology coherent with the compact subsets of that space, so $X\times_k Y$ is the product in the category of $k$-spaces. By $X\times_{CW} Y$ we will denote the CW structure on the product.
Since every CW complex is a $k$-space, being a quotient of a topological sum of balls, which are compact Hausdorff spaces, so is $X \times_{CW} Y$. The projections from $X \times_{CW} Y$ to $X$ and to $Y$ induce the identity map to $X\times Y$, and thus a continuous identity map $i: X \times_{CW} Y \to X\times_k Y$, by universality of the $k$-ification.
In order to show that the inverse $j: X \times_k Y \to X \times_{CW} Y$ is continuous, we show that $j|_K$ is continuous on every compact subset $K$ of $X\times Y$. Since the both projections of $K$ are compact, they are contained in finite subcomplexes $C_X \subset X$ and $C_Y\subset Y$. Their product $C_X\times C_Y$ is compact Hausdorff and thus a $k$-space, and it is a subspace of $X\times_k Y$ containing $K$. Since $C_X \times_{CW} C_Y \to C_X \times C_Y$ is a continuous map from a compact to a Hausdorff space, it is a homeomorphism. That means $j|_K$ embeds $K$ into $C_X\times_{CW} C_Y$ and thus into $X \times_{CW} Y$.
I hope everything is correct, it is easy to lose track with all these different topologies flying around, so I urge you to double check what I did here.