Covering image of a connected CW-complex need not be a CW-complex.

algebraic-topology cw-complexes covering-spaces

Andrew Ranicki conjectured (or at least asked) if a compact, 4 manifold has a CW structure, if and only if, it is smoothable.

As far as I know this is open, but assuming the conjecture is true the answer to your question is that a CW complex may cover a non CW complex. Let $M$ be a nonsmoothable 4-manifold with infinite fundamental group, these are known to exist. Let $\widetilde{M}$ denote its universal cover. $\widetilde{M}$ is noncompact, and it is known that noncompact 4-manifolds are triangulated. Hence, $\widetilde{M}$ has a CW structure and it covers $M$ which is not a CW complex, if Ranicki's conjecture is true.

In this case, the base of this cover is homotopy equivalent to a CW complex since all compact manifolds are homotopy equivalent to CW complexes.

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