Let $G$ be a topological group. Then the classifying space $BG$'s homotopy type depends on the "homotopy type" of the topological group $G$: that is, if $G \to G'$ is a morphism of topological groups (i.e., a continuous homomorphism) which is a weak equivalence, then $BG \to BG'$ is a weak equivalence (note that $BG$ is functorial by the usual construction, and its homotopy groups are those of $G$ with a shift). This means that on a CW complex, giving a principal $G$-bundle is the same as giving a principal $G'$-bundle.
Is there a direct proof of this (e.g. using $H^1$) that does not resort to $BG$?