It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space of diffeomorphisms homotopy equivalent to a CW-complex?

The group $Diff_c(M)$ of compactly supported diffeomorphisms is a nuclear LF space: http://www.mat.univie.ac.at/~michor/manifolds_of_differentiable_mappings.pdf With this topology does it have the homotopy type of a CW complex?