BA are moving from T7 to T8 next month so there won't be CCR for you to try at JFK.
It's not 100% clear what the BA offering for F passengers will be in terms of lounges / facilities wll be in T8 yet.
BA does regular T5 - T3 and vv terminal switches at LHR so it depends on what your onward destination is as to whether you will have to switch terminals or not. Also possible AA may switch some flights back to T5 as well.
If you fly BA F to LHR and your next flight is from T5 then you would have LHR CCR access on the next leg (would need to show your F BP).