There is no airside transit at LHR. You need a boarding pass for that terminal to access the departure area for that terminal and the only way to transfer terminals is via the arrivals stream.

You'll arrive in T3 on AA, and will go into the arrivals passage where you can either go to immigration or flight connections. At flight connections, you will get on a bus to T5 (and will need a T5 boarding pass to get on the bus), which will then drop you off at a corridor leading to security.

The only T3 lounge you would be able to use is the AA Arrivals lounge, assuming it is open when you arrive and you have enough time. You'll have to pass through immigration and enter the UK.
Page 5 - https://www.heathrow.com/file_source...row_T3_Map.pdf
You exit immigration through the world duty free shop labeled 3 on the map. The Arrivals lounge is above the cafes labeled 1 and 2.
After some time in the lounge, you then go to the train station and use the Heathrow Express train to transfer to T5 (Heathrow Express is free between the airport stations).

Arrivals lounges aren't part of the oneworld agreement, so have their own rules for entry. However, OW Emerald having arrived in LHR on an AA flight will get you in.
The only other lounge options you would have are the BA lounges in T5.