I asked the same question in math stackexchange: https://math.stackexchange.com/questions/2322883/how-can-i-endow-a-locally-product-cw-structure-on-a-vector-bundle-over-a-cw-co but it seems that it's harder than I thought, so I ask here:

I'm now learning characteristic classes, and I need a CW structure on the total space of a vector bundle $E\to B$ where $B$ is a CW complex such that the associated sphere bundle and rectriction over any subcomplex of $B$ are both subcomplexes (this is required in "Algebraic Topology from a Homotopical Viewpoint", page 364). I think this should be something like a locally product structure, but I couldn't figure out how to glue them together. I even doubt that this can be done.