Henrik's comment looks like an answer to me.

In odd dimensions you get up to homotopy a sphere and in even dimensions you get something contractible.

Why are we told that $X$ is a CW-complex? If the action of $G$ on $X$ preserves the cells and is free, then surely it will have to be proper.

I'm honoured to be now considered an old timer. I learned a lot of this stuff from Bob Oliver. As you say, Ed Floyd would have been a good source too.

Is the barycentric subdivision of a polytopal triangulation also polytopal? I am not sure, but I suspect that any finite 2-complex will embed in a sufficiently fine subdivision of any triangulation of $S^5$.

It is not so easy to find groups of finite vcd for which $\mathrm{cd}_{\mathbb{Q}}$ is not equal to $\mathrm{vcd}_{\mathbb{Z}}$: I don't know that this is the case for the groups asked about, but it would be surprising if it wasn't.

The mapping telescope $T$ that HJRW mentions is aspherical in both cases, so it is a classifying space for the direct limit group. The asphericity of the mapping telescope doesn't rely on the injectivity, and the dimension of the mapping telescope is one more than the sup of the dimensions of the pieces. (Proof that the mapping telescope is aspherical: any sphere is contained in a finite subtelescope, this finite subtelescope retracts onto its last space and that space is aspherical.)

You're right. Thanks. I'll edit my solution to point out that it has this flaw later.