The most general reference I know for Brown Representability for \emph{cohomology} is Neeman's book on triangulated categories. There even fewer of the Eilenberg-Steenrod axioms are assumed (additivity is not). It's so general that it should cover both cases you list. In this sense, Brown was wrong and his ideas could be made to work more generally, without a countability hypothesis. For homology theories the story is more subtle and countability is often needed. In Neeman's book this theorem is one of the big goals, so it's stated in the introduction and proved close to the end (chapter 12 I believe). Check it out on Google books for a preview

The most general reference I know for Brown Representability for cohomology is Neeman's book on triangulated categories. There even fewer of the Eilenberg-Steenrod axioms are assumed (additivity is not). It's so general that it should cover both cases you list. In this sense, Brown was wrong and his ideas could be made to work more generally, without a countability hypothesis. For homology theories the story is more subtle and countability is often needed. In Neeman's book this theorem is one of the big goals, so it's stated in the introduction and proved close to the end (chapter 12 I believe). Check it out on Google books for a preview