Is there a Van-Kampen-style theorem for universal covers? I was looking for a reference.
I was looking for something like: given two topological spaces $X,Y$, the universal cover of $X\cup Y$ follows from considering (insert a proper construction) applied to the universal covers of $X$ and $Y$.
There is. One reference is Knill, R. J., "The Seifert and Van Kampen Theorem via Regular Covering Spaces", Pacific Journal of Mathematics, 49(1):149-160. Considering the original two universal covers as bundles with discrete structure groups, and with some data about overlap, you can apply the associated bundle and clutching constructions to construct a universal cover of the combined space.
Edit: Maybe an example of a simple case. Suppose that $B_{1}$ and $B_{2}$ are open subspaces of $B = B_{1} \cup B_{2}$, and define $B_{0} = B_{1} \cap B_{2}$. Suppose also that we know the universal covering spaces of all three, regarded as principal bundles with structure groups $\pi_{1}(B_{i})$ given the discrete topology. Denote these principal bundles by $\xi_{i} : E_{i} \to B_{i}$. Now, let $K$ be the pushout of $\pi_{1}(B_{1}) \leftarrow \pi_{1}(B_{0}) \to \pi_{1}(B_{2})$ in the category of groups. Define the associated bundles $\xi_{i} \times_{G} K$, and then define $E = (E_{1} \times_{G} K) \cup (E_{2} \times_{G} K)$ and $\xi = (\xi_{1} \times_{G} K) \cup (\xi_{2} \times_{G} K)$. Then $\xi$ is the pushout of $\xi_{1} \leftarrow \xi_{0} \to \xi_{2}$ in the category of principal bundles. Hence $\xi : E \to B$ has the universal property of the universal cover.
Note: Van Kampen follows from this. The structure group of $\xi$ is $K$, and since $\xi$ is the universal cover we have $K \cong \pi_{1}(B)$. But this means that $\pi_{1}(B)$ is the pushout of $\pi_{1}(B_{1}) \leftarrow \pi_{1}(B_{0}) \to \pi_{1}(B_{1})$, which is what Van Kampen says.