I'm specifically assuming that we have replacement instead of collection; collection breaks things (because then there is a set that contains a map from $n$ to $C$ for every $n\in\mathbb N$, and you can look within that set to get an injection from an infinite subset of $C$ to some infinite set, so $\aleph(C)$ couldn't have been $\aleph_0$.

The construction that seems to work in ZFCA is to take a model of ZFA with infinitely many atoms and then take the direct limit of $L(A_n)$, where $A_n$ is the set of the first $n$ atoms. This seems to be a model of ZFCA, because every set belongs to $L(A_n)$ for some $n$ and therefore inherits a well-ordering from that (as $L(A_n)$ still has a global choice function definable from parameters).

I can't see how replacement fails where collection did, because I can't think any $\varphi$ such that $\forall n\exists!x\varphi(n,x)$ and $\varphi(n,x)$ implies that $x$ is an injection from $n$ into the class of atoms, or anything else that would cause the union of all such $x$s to not fit into any $L(A_n)$.

This of course cannot be extended to a model of GBCA, because the existence of a proper class that is smaller than the universe prevents the existence of a global choice function.

If the construction works and can be extended to ZFC, it does prove that the Hartogs number of classes is interesting in NBG even with the axiom of choice (NBG being conservative over ZFC).

2more comments