I think the standard early reference is the paper by Hotta and Springer here, in which they work with $\ell$-adic cohomology (but the results seem to carry over to other settings). What they show is that the map in cohomology is surjective on the top degree in the target space when mapping into the subspace fixed by the component group, while the precise image in other degrees might be more elusive when there is a nontrivial component group for the unipotent class in question. (This is essentially why all goes well for type $A$, where component groups are trivial.)
Examples in rank 2 for types $B_2, G_2$ show already what goes wrong when the component group is nontrivial (which happens there for the subregular orbit, as Jay observes). But as far as I know, the problem remains quite difficult to study beyond what is done by Hotta-Springer.
[I gave a short survey of Springer theory in Chapter 9 of my 1995 AMS book on conjugacy classes, but unfortunately overstated the Hotta-Springer result in 9.6(4). Corrections are posted on my homepage.]