If you've got bags to check, HKG should book them right through (I'm assuming this is all on one ticket). When you arrive at T3, you can follow the signs to the Flight Connections Centre (bus from T3 to FCC, then walk to T1). This will take you through immigration (and a re-screen for UK security) at LHR, but any checked bags will need to clear customs at EDI.

Alernatively, you can leave any checked bags to go right through & you can enter the UK through T3 to use the AA arrivals lounge. Once you're done, it's a 10 minute walk across to T1 through the network of underground tunnels - You won't be able to go airside in T3 again.