As long as its on one ticket you should have no problems although I would imagine you'll only get the EDI-LHR boarding passes at EDI and willl have to get the AC one at the flight connections centre.

My parents had a similar itinerary NCL-LHR-IAD-CLE with the UK leg on BA and the others on UA. They had to pick up BP's in LHR for UA but with through baggage check

chris