Does the homotopy pullback of a diagram of spaces who are homotopy equivalent to CW-complexes have the homotopy type of a CW complex?

Question

Consider the diagram of compactly generated weakly hausdorff topological spaces:

If each space has the homotopy type of a CW-complex, when does the homotopy pullback of the diagram also have the homotopy type of a CW-complex?

Additionally, if each space is a CW-complex, when does the homotopy pullback of the diagram have the homotopy type of a CW-complex?

Sorry about the poorly formatted image. I don't know how to make tikzcd diagrams in a stack exchange post.

Thanks :)