In Corollary 5.12 of Whitehead's book it is shown that that the cobase change of a homotopy equivalence along a cofibration (NDR pair) is again a homotopy equivalence. This ought to imply homotopy invariance of the double mapping cylinder construction (with respect to homotopy equivalences of the spaces used to form the double mapping cylinder). This is sometimes called the "gluing lemma."
Given a diagram $B \leftarrow A \to C$ of spaces homotopy equivalent to CW complexes Use the map of diagrams $$ |S.B| \leftarrow \quad|S.A| \rightarrow |S.C| $$
$$ \qquad \downarrow \qquad \qquad \qquad \downarrow \qquad \qquad \quad \downarrow \qquad $$
$$ B \qquad \leftarrow \qquad A \qquad \to C $$
where $|S.-|$ in each case means geometric realization of the total singular complex. Then homotopy invariance applied to the above shows that the double mapping cylinder of the top line, which is a CW complex, has the homotopy type of the double mapping cylinder of the bottom line.
As far as finiteness goes, that should follow from Wall's results on finiteness conditions for CW complexes. The point I guess is that the singular chains on the double mapping cylinder is quasi-isomorphic to chain level pushout $C(B) \leftarrow C(A) \to C(C)$ of singular chain complexes, and each of these chain complexes is chain homotopy finite, so the chain level pushout is too.