In Combinatorial Group Theory, Lyndon and Schupp construct a complex $K(X;R)$ from a presentation of group $G=(X;R)$, such that $G \simeq \pi_1(K,v)$ (proposition 2.3, p.117). Moreover, the Cayley complex of $G$ is the universal covering space of $K(X;R)$ (proposition 4.3, p. 124).
The only application I found in the book is the subgroup theorem of Schreier and Nielsen (every subgroup of a free group is free).
Are there any other interesting applications?
These ideas have been developed to higher dimensions, with the intention of constructing resolutions of a group starting from a presentation; the method is to construct inductively the universal covering complex together with a contracting homotopy. The first stage is to choose a maximal tree in a Cayley graph. This has been implemented by Graham Ellis, see https://gap-packages.github.io/hap/www/index.html.
You can also see a groupoid approach to the subgroup theorems, including the Kurosh Theorem, on subgroups of a free product, and a generalisation of Grusko's Theorem, in Philip Higgins' downloadable book Categories and Groupoids.
The relations with topology, and a groupoid, i.e. base point free, approach to covering spaces, are given in my book Topology and Groupoids. The key point in both books is the basic notion of covering morphism of groupoids, and its relation with actions. A virtue of this notion is that a covering map is modelled by a covering morphism, and so lifting of maps is studied via lifting of morphisms, and so is easier to follow and understand.