Here is a quick argument which proves homotopy equivalence directly. First, the pullback of a homotopy equivalence along a Hurewicz fibration is again a homotopy equivalence. Further, by the well-known uniformization theorem of Hurewicz, a locally trivial fibre bundle $E\to Y$ over a paracompact Hausdorff space $Y$ (such as a manifold) is a Hurewicz fibration. We thus conclude that if $f:X\to Y$ is a homotopy equivalence then the map $f^\ast E\to E$ is again a homotopy equivalence.
A few references and remarks
There are many references for the uniformization theorem of Hurewicz, both in articles and in books. For example, in article form, there is the original article by Hurewicz, this article by Dold, and this other article in a slightly more modern language. In book form, there is Spanier's book Algebraic topology, the book A concise course in algebraic topology by May, and tom Dieck's Algebraic topology.
The fact that the pullback of a homotopy equivalence along a Hurewicz fibration is again a homotopy equivalence is called — in the jargon of model categories — the right properness of the Strøm model structure on the category of topological spaces. Actually, any model category in which all objects are fibrant is right proper, and that includes the Strøm model structure. Nevertheless, a direct, elementary proof that the Strøm model structure is right proper can be given using solely the covering homotopy property for Hurewicz fibrations.