Let $T$ be a countable first-order theory, and assume that $T$ has exactly one atomic model up to isomorphism in every uncountable cardinality. (By "atomic" I mean a model which omits the non-principal types).

Now let $\mathfrak{M}$ be a countable transitive model of set theory, and assume that $T$ is also (countable) in $\mathfrak{M}$.

Is the above property preserved in $\mathfrak{M}$? i.e. does it hold in $\mathfrak{M}$ that there exists only one atomic model of $T$ in every uncountable cardinality in $\mathfrak{M}$?