I think the relevant formula is 8.4.1 in T&G. This is applied in section 9.2 to the Phragmen-Brouwer property and the Jordan Curve Theorem.
My original motivation for the investigation was to avoid a detour to compute the fundamental group of the circle: a basic theorem should compute THE basic example! I like the view of the integers (an infinite set) as an identification of a groupoid $\mathbf I $ with 4 arrows, identifying 0 and 1.
Also I tend to see covering spaces in terms of covering morphisms of groupoids, since then a covering map is algebraically modelled by a covering morphism, whereas an action is one step further.
In the new book `Nonabelian algebraic topology', published by the EMS, the van Kampen style arguments are used to compute relative homotopy groups as modules, and second relative homotopy groups as crossed modules, using colimit calculations.
January 25,2016 There is a small correction to the Phragmen-Brouwer proof.
It may be useful to think about forming the coequaliser of two morphisms $a,b: G \to H$ of groupoids. If $a,b$ are the identity on objects, the formula is just as you would expect: factorise by the normal subgroupoid of $H$ generated by ....If they are not the identity on objects, you first have to coequalise the objects. This gives a function $f: Ob(G) \to Ob(H)$. So you now need Higgins' "Universal morphism" say $U_f : G \to f_*(G)$. The construction of this generalises free groups, free products of groups, free groupoids ... and is well explained in Higgins' book. "Categories and Groupoids" available as a TAC reprint.
Nov 3, 2020 In the NAT book, Appendix B, this is put in the context of the functor $Ob: Gpds \to Sets$ being a bifibration of categories.