The elements of sets in the theory ETCS are not other sets, they are functions $\ast \to X$. In particular, elements to not themselves have elements, so you cannot discuss the union as you have described it. In essence, this is one of the main difference between ETCS (a _structural_ set theory) and ZF(C) (a _material_ set theory).

The translation from ETCS is not simplistic. One has to take certain well-founded trees where the nodes are ETCS-sets to recover ZF(C)-sets. Actually, since ETCS is really of the strength of bounded Zermelo set theory with Choice you get this instead (see <http://ncatlab.org/nlab/show/pure+set>). Such a tree is actually a _diagram_ $\mathcal{T} \to Set$ where $Set$ is given by ETCS.

Then given a rooted tree (with root $T_0$) representing a material set, the union is the disjoint union at each level of the forest of rooted trees gotten by chopping off $T_0$.