homotopy theory - Explicit classifying spaces for crossed complexes

It is not clear to me what you need / want. The classifying space of a cyclic group is constructed using a presentation and then killing off higher identities that may be around (there aren't any!). From that viewpoint the question you seem to ask is related to the combinatorial group theory of the group in question (or am I misderstanding the question.) There are examples that might help due to Loday in his paper on higher syzygies, but that may not quite fit the bill as he does not explicitly give the link with classifying spaces.

If you are happy with simplicial methods then you can build a simplicial T-complex from a crossed complex of groups by a modified Dold-Kan construction. The classifying space of that simplicial group (its Wbar) is something that has the same properties as Ronnie's classifying space. It is feasible if you know the crossed complex reasonably fully to construct this explicitly.

It depends what you need the construction for? How is your crossed complex arising precisely (and incidently what do you mean by the `topology', will a simplicial model do)?