See The smallest singular value of deformed random rectangular matrices.
If the diagonal elements of $D$ are of order unity then the smallest singular value of $DS$ is of order $\sqrt{m}-\sqrt{n}$ with high probability. The largest singular value is of order $\sqrt{m}+\sqrt{n}$.
If there is no constraint on $D$ the upper bound is the product of the largest singular value of $D$ and the largest singular value of $S$ (the latter being of order $\sqrt{m}+\sqrt{n}$).