Changing planes in Heathrow

Hardy perennial question. Note Customs and Immigration are two different things. Customs = Goods, Immigration = people (passport checks)
Is it one through booking? Sounds like yes. In which case the following applies:

• Your luggage will be labelled through to your final destination (DFW). You will not see it until there, where you will pick it up and take it through customs.
• You will go through exit passport check (emigration) at Copenhagen. If you have a non-EU passport it will be stamped there for exiting the Schengen Area.
• At Heathrow you will stay "airside", no immigration or emigration.
• When you land at your final destination, you will go through immigration (passport check).

American doesn't fly from Heathrow to Copenhagen - I would imagine this is a codeshare on British Airways.

If you have a proper through booking on a single ticket then there is no need to go through customs & immigration or collect bags. The latter are through-checked and you make a transfer airside.

Outward this would be terminal 3 (AA) -> terminal 5 (AA/BA).
Inward this would be terminal 2 (SK)-> terminal 3 (AA).

Just follow the connections signs.

If you aren't on a single ticket (and some third party booking sites can be naughty this way) then you would need to collect checked bags as they won't be through-checked.

Go to this website: Heathrow Airport Flight Connections plug in your flight information and it will give you step by step instructions.

Also look to see if your flights are really on American or codeshares with British Airways operating the flights. It should say so in your reservation.

You are right. The flight from DFW to Heathrow is AA, but is AA operated by British Air to Copenhagen. They are all booked through AA so I should be all right. On the way back it is SAS then AA. Great information, Thanks!