For an example of a super-stable theory failing the property, you can take infinitely many unary predicates (so $Th(2^\omega, U_n: n \in \omega)$ where $U_n(\eta)$ holds iff $\eta(n) = 1$). Then if $\nu$ and $\mu$ satisfy that if $\nu[G_0]$ embeds into $\mu[G_1]$ in $\mathbb{V}[G_0 \times G_1]$ then every quantifier-free type realized in $\nu[G_0]$ is also realized in $\mu[G_1]$, hence is in $\mathbb{V}[G_0] \cap \mathbb{V}[G_1] = \mathbb{V}$. So let $\mathbb{P}$ adjoin a Cohen real $x$ and let $\nu$ be a name for the model of $T$ corresponding to $\{\eta: \mbox{supp}(\eta) \mbox{ is finite}\} \cup \{x\}$.
I can give an exact characterization if we replace homomorphism with elementary embedding (which isn't much of a loss, since we can just Morleyize) and require $T$ to be complete, in a countable language:
Say that $T$ is generically elementarily embeddable (g.e.e.) if for all $\mathbb{P}$-names $\nu$ for a model of $T$ there is a $\mathbb{Q}$-name $\mu$ for a model of $T$ such that $\mathbb{P} \times \mathbb{Q} \Vdash \nu[G_0]$ elementarily embeds into $\mu[G_1]$.
Claim. If $T$ is a complete theory in a countable language, then $T$ is g.e.e. iff $T$ is small.
Proof. It follows from the previous example essentially that g.e.e. implies $T$ is small. (If $T$ is not small we can arrange $\nu \models T$ realizes a type not in $\mathbb{V}$, which is enough.)
For the reverse direction, let $\mathbb{P}, \nu$ be given. Let $\mathbb{Q}$ force that $\nu$ (i.e., the name in $\mathbb{V}$) becomes countable and let $\mu$ be a name for a countable saturated model of $T$. Then in $\mathbb{V}[G_0 \times G_1]$, $\nu[G_0]$ is countable, $\mu[G_1]$ is countable and saturated (since being $\aleph_0$-saturated is absolute for small theories) and so $\nu[G_0]$ elementarily embeds into $\mu[G_1]$.
P.S. Chris Laskowski, Richard Rast and myself independently investigated several similar things to the papers you quoted, see A New Notion of Cardinality for Countable First Order Theories