This is not an answer but I think points to a line of investigation.

We know for a CW-complex $X$ that $X$ is the colimit of its skeleta $X^n$, and that $X^{n+1}$ is obtained from $X^n$ by attaching cells, which again is a colimit operation.

I now refer to the general theory of fibred exponential laws, which have been investigated over many years by Peter Booth, also with colleagues at Memorial University, and we wrote two papers:

(1) Spaces of partial maps, fibred mapping
spaces and the compact-open topology, Gen. Top. Appl. 8
(1978) 181-195.

(2) On the application of fibred mapping
spaces to exponential laws for bundles, ex-spaces and other
categories of maps, Gen. Top. Appl. (1978) 165-179.

both available from my publications list as [26,27], and from the Journal's web page. These papers show that under suitable restrictions, or by working in a convenient category of spaces, see Section 7 of (1), and Section 8 of (2), that if $p:Y \to X$ is a map of spaces then the pullback functor $p^*: Top/X \to Top/Y$ has a right adjoint and so **preserves colimits**. So if $Y^n= p^{-1}X^n$ you get that $Y$ is the colimit of the $Y^n$.

Now you also have a chance of analysing the construction of $Y^{n+1}$ from $Y^n$. I think this easily gives the case mentioned by Fernando, when $p$ is a covering: that result goes back to JHCW, with a direct proof.

You need to look carefully at the pullbacks of the elements of the construction of $X^{n+1}$. Perhaps you also need to look at generalisations of CW-complexes.
See for example Minian, G.; Ottina, M. A geometric decomposition of spaces into cells of different types. J. Homotopy Relat. Struct. 1 (2006) 245–271 and its sequel.