Short answer for the simplicial case: yes.

Let X• be a simplicial object in finite CW complexes. Then there is a simplicial object K• in finite simplicial sets and a zig–zag of levelwise weak equivalences of simplicial objects |K•| ~→ |Sing X•| ~→ X• so that, in particular, each |Kn| has the homotopy type of Xn and the maps respect the simplicial structure (strictly) in the middle object.

Here is a clean way to see this.

1. Replace Top by sSet. The Quillen adjunction |–| ⊣ Sing between simplicial sets and compactly generated spaces gives natural weak equivalences |Sing Xn| → Xn for CW complexes Xn, levelwise and natural in the simplicial direction. Thus it suffices to replace the strict simplicial diagram Sing X• in sSet by a levelwise finite simplicial set diagram up to levelwise weak equivalence.

2. Choose finite models levelwise. For each n, pick a finite simplicial set Kn together with a weak equivalence Kn → Sing Xn (e.g. via a chosen finite triangulation of Xn and the unit Kn → Sing|Kn| composed with |Kn| → Xn and Sing). This gives you the objects you want in each degree, and levelwise weak equivalences αn: Kn → Sing Xn.

3. Transport the simplicial structure up to coherent homotopy. Using the strict simplicial structure on Sing X• and the αn, you obtain a homotopy coherent simplicial diagram on the objects Kn (i.e. the faces and degeneracies are defined up to higher coherent homotopies so that α•: K• ⇝ Sing X• is a homotopy coherent natural transformation).

4. Rigidify (rectify) the homotopy coherent diagram. By the rigidification (or rectification) theorem for homotopy coherent diagrams in a simplicial model category (Top and sSet both qualify), any homotopy coherent Δop-diagram is weakly equivalent to a strict Δop-diagram, and one can do this without changing the objects. Concretely, there exists a strict simplicial object K′• in sSet with K′n = Kn for all n, together with a map of simplicial objects K′• → Sing X• that is a levelwise weak equivalence. References for this rigidification include:

• H. Vogt, Boardman–Vogt W-construction (rigidification of homotopy coherent diagrams),
• Dwyer–Kan, Function complexes in homotopical algebra,
• Cordier–Porter, Homotopy coherent diagrams and their realizations,
• Lurie, Higher Topos Theory, Proposition 4.2.4.4 and Corollary 4.2.4.8 (expressing functor ∞-categories via projective model structures and giving rectification).

5. Realize levelwise. Since all simplicial sets are cofibrant and realization is left Quillen, the levelwise weak equivalence K′• → Sing X• becomes a levelwise weak equivalence |K′•| → |Sing X•|. Composing with the natural weak equivalence |Sing X•| → X• yields the desired zig–zag |K′•| ~→ |Sing X•| ~→ X•. Because K′n = Kn are finite simplicial sets, we have produced the required simplicial object of finite simplicial sets.

Remarks.

• The obstruction Tyler alluded to for arbitrary diagrams is circumvented here because we do not try to make a functorial triangulation on the nose; instead we pass through Sing and use rigidification of homotopy coherent diagrams to obtain a strict simplicial diagram in finite simplicial sets up to levelwise weak equivalence.
• The directions of the arrows in the zig–zag are immaterial; the question only asks for a zig–zag of levelwise weak equivalences connecting the given simplicial CW diagram to the levelwise realization of a simplicial diagram of finite simplicial sets.

Generalization to CWf(A). Let C be a simplicial, left proper, combinatorial model category, and let A be a set of cofibrant homotopy-compact objects. Let CWf(A) denote the finite A-cell objects (obtained from A by finitely many homotopy pushouts and finite coproducts), and let S•f(A) be the finite simplicial objects built from finite coproducts of A, degenerate above some dimension. Then:

• Every object of CWf(A) is (canonically up to weak equivalence) the geometric realization of some object of S•f(A) (build a finite simplicial resolution by cells; this is a two-sided bar construction/Segal-type presentation).
• Given a simplicial diagram X• in CWf(A), carry out the same three-step procedure:
  1. choose levelwise presentations Xn ≃ |Kn| with Kn ∈ S•f(A),
  2. transport the simplicial structure to a homotopy coherent diagram on the Kn,
  3. rigidify in the simplicial model category C to get a strict simplicial object K′• ∈ S•f(A) together with a levelwise weak equivalence |K′•| → X•.

Thus, under these mild hypotheses on C and A, the answer to the generalized question is also yes.

In ∞-categorical terms: the relative categories (FinCW, we) and (FinSSet, we) present the same small ∞-category of finite homotopy types. Applying Fun(Δop, –) preserves equivalences of ∞-categories, and every ∞-functor from Δop can be rectified to a strict functor in a simplicial model presenting the target. This is exactly the mechanism used above.