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 $T_0$.