Is $2^{1+\sqrt{mn}}$ an upper bound for the size of the DFA resulting from the determinisation of n DFAs of size m
No.  Asymptotically, the size can be as large as $\sim m^n$.

Let $p_1,\dots,p_n$ denote the $n$ largest prime numbers that are at most $m$.  Let

$$L_i = \{x : x \not\equiv 0 \pmod{p_i}\},$$

where we assume the number $x$ is expressed in unary.  Then $L = L_1 \cup \dots \cup L_n$ has the form

$$L = \{x : x \not\equiv 0 \pmod q\},$$

where $q = p_1 \times \dots \times p_n$.

Note that, with this construction, each $L_i$ has a DFA of size at most $m$ (in particular, of size $p_i$).  However, the minimal DFA for $L$ has size $q$, which is close to $m^n$.  In particular, the size of the minimal DFA for $L$ is asymptotically larger than $2^{1 + \sqrt{mn}}$ (as a function of $n$, treating $m$ as fixed).