Uniqueness of Chern/Stiefel-Whitney Classes

I'm going to assume that your characteristic classes are supposed to live in the singular cohomology of the base space. Then to show your uniqueness result, it should be enough if you can produce, for any space $B$, a map $f:B'\to B$ such that $B'$ is paracompact, and $f$ induces an isomorphism in singular cohomology.

For any $B$, you can find a CW-complex $B'$ and a weak equivalence $f: B'\to B$, by one of Whitehead's many theorems. Weak equivalences always induce isomorphisms in singular cohomology, and CW-complexes are paracompact (I think!). Hatcher's topology textbook proves all of these, except possibly for the paracompactness claim (for which I can't find a reference yet).

If you're talking about Cech cohomology, then this proof won't work.

You do not need to have metric to have splitting principle. Let $p: E \to B$ be a $n$-dimensional vector bundle and as usual let $E_0$ consist of nonzero vectors. Then consider the $\bar{E}= E_0/ \sim$ where $\sim$ identifies any two vectors in the fiber if they lie on the same line passing through origin(which just requires vector space structure)(Milnor and Stasheff describe this as defining fiber of new bundle as the quotient vector space $F/(\mathbb{C}v), v \neq 0$ pg157). Then $\bar{p}: \bar{E} \to B$ defines the associated projective bundle. We obtain splitting map by repeating this $n$-times.

But the problem with non-paracompact space $B$ is that it may not admit a classifying map $f : B \to Gr_n$ such that $f^{\ast}\gamma^n \cong E$(bundle isomorphism) where $p : E \to B$ is $n$-plane bundle. Since for a given cover there is no locally finite refinement, in the local trivializations of vector bundle $p: E \to B$ we cannot use the Urysohn lemma properly to get a bundle map $f' :E \to E(\gamma^n)$ which restricts to linear map in each fiber as Milnor-Stasheff do in the proof of lemma 5.3. So we cannot prove theorem 5.6(Milnor-Stasheff) for non-paracompact spaces and cannot define $w_1,c_1$(characteristic classes) as they define in page 69.

In the case of long line there is a paper showing that its tangent bundle is nontrivial without using any characteristic classes.