It is well-known that for any finite CW complex $K$ one can construct a finite simplicial set whose geometric realization is equivalent to $K$. But the construction is not strictly functorial.

Suppose we have a diagram of finite CW complexes. Can one always construct a diagram of finite simplicial sets, whose geometric realization is connected to the original diagram by a zig-zag of natural weak equivalences?

One may ask a similar question in a context more general than the category of spaces. Suppose we have an $\infty$-category $\mathcal C$. Suppose $A$ is a collection of (say compact) objects of $\mathcal C$. Let $CW^f(A)$ be the category of finite cellular objects generated by $A$. Roughly speaking, $CW^f(A)$ consists of objects that can be built as a finite homotopy colimit of objects of $A$. I am leaving the definition of $CW^f(A)$ a little vague - I am happy to use any reasonable notion. Let $SS^f(A)$ be the category of simplicial objects in $\mathcal C$ that in each simplicial degree are a finite sum of elements of $A$, and are degenerate above some dimension. Geometric realization gives a functor $SS^f(A)\to CW^f(A)$. Now we have an obvious generalization of the question:

Suppose we have a diagram in $CW^f(A)$. Can we find a diagram in $SS^f(A)$ whose geometric realization is (weakly) equivalent to it? If it is not true in general, is there a reasonable set of sufficient assumptions?