complexity of outputting the union of a collection of subsets of a set
That's not possible.  Any correct algorithm will have to read all of the input.  The input size is $\Theta(mk)$, so that means any correct algorithm will have to have running time at least $\Omega(mk)$.  $O(m+k)$ and $O(m+k) p(\log(m+k)))$ are not achievable.