AA will typically check across tickets for other OW carriers. But, it cannot check other than along the same route as you have ticketed. And it cannot check through the same station twice. You will face that LHR problem twice on this itinerary.

Thus, in this case, the furthest AA will check is the JFK-LHR segment. At LHR you would need to reclaim, check to CPH and then at CPH check through to EDI.