Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex? Does a compact semilocally simply connected geodesic space have the homotopy type of a compact CW complex? Actually what I'd like to know is whether the fundamental group of such a space is finitely presented. 
Edit: As Bruno Martelli notes, this is obviously false, but the question of whether the fundamental group is finitely presented, which is what I really want to know, is still open.
 A: I hope I have not goofed, but I think the answer to your modified question is yes:

The fundamental group of a semi-locally simply connected, compact and geodesic space is finitely presented.

Here are the ingredients - all numbers in parentheses refer to Bridson-Haefliger, Metric spaces of non-positive curvature, Springer Grundlehren, 1999, Part I.


*

*The universal covering space $\widetilde{X}$ equipped with the length metric induced by the covering projection is a length space (3.25).

*The fundamental group $\pi_{1}(X)$ acts on $\widetilde{X}$ properly and cocompactly by isometries (8.3 (2)), see also (8.5).

*If a length space $\widetilde{X}$ admits a proper and cocompact action by isometries then it is locally compact (8.4 (1)) and hence proper and geodesic (3.7).

*A group is finitely presented if and only if it acts properly and cocompactly by isometries on a simply connected geodesic space (8.11).


All this taken together yields that $\widetilde{X}$ is a simply connected geodesic metric space and $\pi_{1}{(X)}$ acts properly and cocompactly by isometries, hence $\pi_{1}{(X)}$ must be finitely presented.
A: The bouquet of infinitely many shrinking 2-spheres in $\mathbb R^3$ centered in $(0,0,n)$ and of radius $n$ is a compact simply connected geodesic space, which is not homotopic to a compact CW complex.
A: In Bourbaki's Topologie Algébrique text, a space is called délaçable*  if it is locally path-connected and semi-locally simply-connected.
Theorem 1 in chapter IV, §2, No.3 states that

Let $X$ be a compact délaçable space and $x \in X$. Then $\pi_1(X,x)$ is finitely presented.

*I have seen unloopable for its English but that does not seem like a common use.
