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, and represents a membership tree.

Then given a rooted tree (with root $T_0$) representing a material set, the union is (roughly) the tree with root the disjoint union of sets at the next level up from the original root and the union of all the branches above that.