We have an algebraic classification of WZW terms that works for any dimensions and for any symmetry groups (including discrete groups). In fact what we did is that we classified the so call SPT states for any on-site symmetries in any dimensions. The boundary excitations of a SPT states are describe by effective non-linear sigma-model of the symmetry group with a WZW term. The classification of the bulk SPT states, in turn, classify the WZW terms for the effective boundary non-linear sigma-model.
The classification can be stated as the following (arXiv:1106.4772):
Consider a non-linear sigma-model whose target space is the symmetry group $G$ in $d$ space-time dimensions, its WZW terms are classified by group-cohomology classes $H^{d+1}(G,R/Z)$. Here $G$ can be continuous or discrete.
Add: What we really classified is the WZW term in a non-linear sigma-model whose target space is the symmetry group $G$ in $d$ dimensional space-time lattice. That is we have a discrete space-time. Even for discrete space-time and discrete groups, a generalization of WZW term can be defined, and their classification is purely algebraic. This is why our classification involves co-homology theory rather than homotopy theory.
If we break the symmetry, the low energy effects of WZW term disappear (ie the low energy properties of the theory is the same with or without WZW term. This is why we require symmetry.