Question about a semi-decidable property of languages
Let $M$ be any computably enumerable set and define the set $$J_M = \lbrace n \in \mathbb{N} \mid W_n \cap M \neq \emptyset\rbrace,$$
where $W_n$ is the $n$-th computably enumerable set. Then $J_M$ is a computably enumerable set because
$$m \in J_m \iff W_n \cap M \neq \emptyset \iff \exists m . m \in W_n \land m \in M$$
is obviously a semidecidable condition, as c.e. sets are closed under finite intersections and  existential quantification over $\mathbb{N}$.

We may now get your result by taking $M = L_{\mathrm{uni}}$.

If you are wondering about the "obviously semidecidable" condition, we can put in a bit more details: to semidecide whether $W_n \cap M \neq \emptyset$ just keep enumerating the elements of $W_n$ and $M$ in parallel, until an $m$ appears in both enumerations.