Suppose $\mathcal L \subseteq \mathcal L’$ are first-order languages, $\kappa$ is a cardinal, and $T’$ is a theory in $\mathcal L’$ that is $\kappa$-categorical. Let $T = T’ \restriction \mathcal L$. Is $T$ $\kappa$-categorical?
If $|\mathcal L’| = \kappa = \aleph_0$, then I can show the answer is yes using the Ryll-Nardzewski Theorem, which says that a countable theory is $\aleph_0$-categorical iff for each $n$, the number of $n$-types is finite.
The answer is no, one can lose categoricity in a reduct of a theory. Consider the following example.
Consider the theory $T$ describing a bijection between two disjoint infinite predicates $f:A\to B$. So a model consists of two disjoint parts, the $A$-part and the $B$-part, and a bijection $f$ between them. The language is $\{f,A,B\}$.
This theory is categorical in every cardinality. But if we restrict the theory to its consequences in the language with the two predicates $\{A,B\}$ and without the bijection, then it is just the theory of two disjoint infinite predicates, which is no longer categorical in uncountable powers, since one predicate could have a different cardinality than the other.
One can also break countable categoricity when $|\mathcal{L}'|>{\aleph_0}$. This example comes from an undergraduate course I took with Malliaris.
Let $\mathbb{P}$ be the collection of primes and let $\mathcal{P}(\mathbb{P})$ be the powerset of $\mathbb{P}$. Let $\mathcal{L'} = (+,\times,0,1;(D_{\alpha}(x))_{\alpha \in \mathcal{P}(\mathbb{P})})$ and let $T \models Th_{\mathcal{L}'}(\mathbb{N})$ with the usual interpretation of the symbols, and for each $\alpha \in \mathcal{P}(\mathbb{P})$, $\models D_{\alpha}(n)$ if and only if $n \in \alpha$. One can show that $T$ is $\aleph_0$-categorical and that the only countable model is the standard model. This follows from the fact that if one adds a single non-standard element, one must necessarily add $2^{\aleph_0}$ many elements.
However, if we let $\mathcal{L} = \{+\}$, then by Ryll-Nardzewski, we no longer have countable categoricity.
