at.algebraic topology - Universal Covering Space of Wedge Products

Today I was studying for a qualifying exam, and I came up with the following question;

Is there a simple description in terms of the subspaces universal covers for the universal cover of a wedge product?

This question came about after calculating universal covers of the wedge of spheres ($mathbb{S}^1 veemathbb{S}^1$ and $mathbb{S}^1 veemathbb{S}^n$) and the wedge of projective space with spheres. In these cases, the universal cover looks like the cross product of the sheets of the universal covers of each space in the wedge.

For the case of wedging two spheres, we can use the fact that $pi_{ngeq2}left(Uright)$ is isomorphic to $pi_{ngeq2}left(Xright)$ for $U$ covering $X$.

I googled around a bit to try and find something, but nothing appeared.

Thanks in advance!