The universal cover of a graph of groups is the Bass--Serre tree. This is described in Serre's book Trees, to which you refer. I don't have a copy to hand, so I can't give you the precise reference, but it's the main object of study throughout the book.
Let me also add that, when trying to think about coverings of graphs of groups, I find it much easier to think about coverings of graphs of spaces. The initial reference for this is paper of Scott and Wall. My students wrote some things about this point of view in this blog.
That said, it's not always feasible to describe every finite-sheeted covering of a graph of groups. For instance, in the case that you ask for, the set of coverings that you ask for is in bijection with the set of transitive permutation groups generated by a pair of permutations of order $l$ and $m$. Just the set of regular coverings is in bijection with the set of groups $\Gamma$ equipped with a generating set $\{a,b\}$, where $a$ is of order $l$ and $b$ is of order $m$. This is an unimaginably complicated set!