It seems that at least a partial answer can be given using the formalism of $\infty$-topoi: if $X$ is a paracompact Hausdorff space which is locally contractible (in the strong sense discussed above) and such that the $\infty$-topos $Sh(X)$ of sheaves (of $\infty$-groupoids) on $X$ is hypercomplete, then $X$ is homotopy equivalent to a CW complex. This happens, for example, when $X$ has finite covering dimension.
To see this, recall Lurie's HTT section 7.1. Given a suitable basis $B$ for the topology on $X$, Lurie constructs a model structure on the category $Top_{/X}$ of spaces over $X$ which is a model for the $\infty$-category $Sh(X)$. If $X$ is locally contractible then one can construct a hypercovering of the terminal sheaf on $X$, given concretely by a split simplicial object $U_\bullet$ in $Top_{/X}$ such that each $U_n$ is a coproduct of contractible open subsets of $X$. The geometric realization $|U_\bullet| \in Top_{/X}$ encodes a sheaf which is $\infty$-connective. When $Sh(X)$ is hypercomplete the sheaf $|U_\bullet|$ admits a section (which in this case will be realized by an actual map $X \to |U_\bullet|$ in $Top_{/X}$ sensesince in this model structure the terminal object is cofibrant and every object is fibrant) and so $X$ is a retract of $|U_\bullet|$. Since $U_\bullet$ is a split simplicial object the realization $|U_\bullet|$ coincides with the corresponding homotopy colimit, and since each $U_n$ is contractible this means that $|U_\bullet|$ is homotopy equivalent to a CW complex. Since $X$ is a retract of $|U_\bullet|$ it follows that $X$ is homotopy equivalent to CW complex as well.
This answer is not unsatisfying, but it would be more satisfying if the assumptions above were met by every space which is actually a CW complex. This leads to the following natural question: is $Sh(X)$ hypercomplete for every CW complex $X$?
