Theorem 4.7 in the BRICS version of Lack and Sobicisnki's Adhesive categories shows that any adhesive category has the "effective unions" (in Barr's terminology) of the question. However, if you look at the proof, you will find that the full strength of adhesivity is not used. Unwinding the proof, we need substantially less:
Theorem: (Lack and Sobicinski): Let $\mathcal C$ be a category. Suppose that
- $\mathcal C$ has pullbacks along monomorphisms;
- $\mathcal C$ has pushouts of monomorphisms along monomorphisms, and the resulting pushout squares consist of monomorphisms;
- Whenever $A \xrightarrow f C \xleftarrow g B$ is a jointly epimorphic pair of monomorphisms in $\mathcal C$, and $C' \xrightarrow h C$ is any map, then the base change $A' \xrightarrow{f'} C' \xleftarrow{g'} B'$ is a jointly epimorphic pair of morphisms.
Then $\mathcal C$ has effective unions.
Notes:
(3) holds if $\mathcal C$ has binary coproducts and pullback-stable epimorphisms.
In particular, this theorem implies both that topoi have effective unions and that abelian categories have effective unions.
To be explicit: following Barr, I say here that $\mathcal C$ has effective unions if the subobject posets in $\mathcal C$ have binary joins, which are given by $A \cup B = AB = A \amalg_{A \cap B} B$ as in the question.
I think this still doesn't cover the abelian case.