Upper bound for the union of $n$ DFAs of size at most $m$?
Yes.  There is a simple proof.

You say that $L_i$ are recognized by DFAs of size $\le m$, and that there are at most $f(m)$ possible such DFAs.  Therefore, there are at most $f(m)$ possible such languages.  If $n \ge f(m)$, then $L_1,\dots,L_n$ must contain some repetitions (some language appears multiple times).  Remove all repeated instances, so the $L_i$ are unique.  Then at that point we have $n \le f(m)$.  In other words, without loss of generality we can assume $n \le f(m)$.

Now the union of $n$ such languages can be recognized by a DFA of size at most $m^n$ (by a product construction), and $n \le f(m)$, so the union of $n$ such languages can be recognized by a DFA of size at most $m^{f(m)}$.