BAA or HM Someone does not permit ARRIVING passengers without ongoing connections access to the airline lounges at LHR, reagrdless of class or status. You can access the AA Arrivals Lounge, howeever, if you arrive during open hours and are willing to purcha$e the admit.

THe LHR Arrivals Lounge can be a bit tricky to find, but it's a decent shower and rest after a flight (but iirc, it closes about 14:00.) You will be able to acce$$ the lounge...

Per aa.com:

"Arrivals Lounge

All First and Business Class passengers flying American to London Heathrow are invited to enjoy the Arrivals Lounge amenities upon their arrival (Economy Class passengers can pay a fee of $35 USD for access to the London Heathrow lounge upon arriving). To enter, simply show your boarding pass at the front desk.

The Arrivals Lounge offers services and amenities, such as:

20 Private showers
Hair dryers
Trouser presses
Complimentary beverages
Cappuccino/Espresso
Breakfast bar
Mini gym
Fax/copy facilities
Pay/credit card phones
Individual work stations

The London Heathrow Arrivals Lounge can be found in Terminal 3. Exit to the right after Terminal 3 Customs Hall, then take the stairs or elevator to the Mezzanine level."