Let $G$ and $H$ be two topological groupoids. Suppose that I have two morphisms $G \rightrightarrows H$ and I want to take the 2-coequalizer of these maps. I'd like an explicit description of (a particular model for) this weak colimit. I can do this very easily for groupoids in SET by constructing a groupoid with objects $G_0 \coprod H_0$ where I desribe the arrows in terms of generators and relations. However, then I don't know what topology to put on the arrows. I'd also be happy, if instead of this, someone knew an explicit description of a weak pushout diagram of topological groupoids.
General colimits of topological groupoids are shown to exist in the paper with Lew Hardy referred to by Jeremy (Math. Nachr. 71 (1976) 273-286.); essentiall,y existence is an application of the Freyd adjoint functor theorem. This also makes it quite difficult to describe the topology explicitly, but in practice we often want only the universal property.
Homotopy colimits are constructed as colimits: for example a homotopy pushout is a double mapping cylinder, where the topologically discrete groupoid $\mathcal I$, the groupoid version of the unit interval, with two objects $0,1$, takes the place of the usual unit interval. The general homotopy colimit is more complicated, and I think you would have to refer to specialised papers, or ask Tim Porter!