I search for a chain of clean references, which lead the fact of topological manifolds of dimension $n$ having the homotopy type of a CW of dimension $n$.

Milnor's On spaces having the homotopy type of a CW-Complex proves that every topological manifold has the homotopy type of a countable CW-complex since it is an absolute neighborhood retract.

Theorem E of Wall's Finiteness Conditions for CW-complexes then gives me the desired answer for dimensions at least 3, as long as I know that the universal covering $\tilde{M}$ of a topological manifold $M$ of dimension $n$ has vanishing homology up to dimension $n$ and it holds $H^{n+1}(M,\mathcal{B})=0$ for all abelian coefficient bundles $\mathcal{B}$.

Theorem 3.26 and Proposition 3.29 of Hatcher's book gives me the first claim about the homology since universal coverings of topological $n$ manifolds are again topological $n$ manifolds since the fundamental group of a topological manifold is countable (a clean reference for this is Theorem 7.21 in Lee's book 'Introduction to Topological Manifolds')

Since I am happy to forget about the low dimensional cases, it leaves me with the search for a solid reference to the claim

Let $M$ be a topological manifold and $\mathcal{B}$ be an abelian coefficent bundle, then $H^{n+1}(M,\mathcal{B})=0$.

finiteCW-complex? Then, what do you call a manifold? $\endgroup$