double flight number--anybody ever encounter this?

It sounds like you're describing codesharing.

https://en.wikipedia.org/wiki/Codeshare_agreement

When you look at the departure board, you'll often see a single flight cycle through two or more flight numbers for the different airlines. Flights are ordered by destination and then by departure time. So you find your destination and departure time and you'll see your flight, and if you watch, the airline and flight number will change periodically. It's extremely common.

What you're describing is a code-sharing partnership between AA and BA ...

Usually flight numbers are three digits or less. If the flight number has four digits and starts with the number 3 or higher, it’s probably a code-share flight being operated by another airline.

Happens all the time when airlines share routes and planes. I’ve been on flights with 5 different flight numbers, each airline having its own flight number. Using your case as an example, the outbound plane may be British air while the Inbound would be an American. United teams a lot with Lufthansa. On the departure board, it will cycle through the flight numbers, all with the same gate and departure time. Whichever airline you bought your ticket through will be the flight number on your boarding pass.

This is common. In fact, these days it's unusual to get a flight that is NOT a codeshare.

When you're in the airport, take a look at the digital signs for all the flights (arrivals/departures). You will notice that a single flight may have 3, 4, 5 or more different codeshares - you'll see the same flight into (gate, time, departure) remaining the same, but the airline and flight number cycles through multiple carriers. Not unusual. At all.

The USDOT requires that the difference between marketing carrier (the airline that sold you the ticket) and operating carrier (the airline that flies that particular flight) be labeled at the time of purchase for US airlines and flights that touch US soil. This rules covers both the typical codeshares as well as an airline's regional partners. (I'm looking at a route now where it's noted that "DL 5293 is operated by Endeavor Air DBA Delta Connection" or "DL 5894 is operated by Republic Airline DBA Delta Connection" for the final leg)

So if there's a particular airline you like to avoid, it's good to look for any of that fine print.

But in coming home to California, would the Gatwick flight list show the BA flt number, the American Flt number, or BOTH-- and would you report to the BA terminal counter, or American?

In the lingo of the frequent flyers who populate the Air Travel boards like TripAdvisor and Flyertalk, you will be flying on BA "metal" although you apparently booked with American Airlines.

When you look at the Departures board at the Oakland Airport, you may see other airlines such as Finnair or Iberia listed for the same flight. Since they list the flights alphabetical order by deatinTion, just look for "London."

Your flight is a 777. I am curious what your options for advance seat selection are, since BA charges for that.

You would check-in for your flights with the airline that you purchased your ticket from.

No, you check-in with the airline whose plane you are actually flying in--in the OP"s case this will be BA. On my first European vacation, I ran into exactly this--an itinerary that was sold by Lufthansa, whose first segment (with a Lufthansa flight number) was actually a United flight to Chicago. We went to Lufthansa's check-in area at LAX and were sent to the United counters (which, since this was LAX, was in a completely different terminal a fair walk away.) Lesson learned.

As far as what flight number will appear on the terminal board, it will probably be both--there might be two (or more) separate listings or the Airline/flight number combination will "roll over" on the display, with all the other info (departure time, destination) staying the same.

It is quite possible that the flight will have an Iberia number and a Finnair one too. They will rotate on the departure board.

Look at your itinerary, it will have which airline you will be flying on. In September my friend who lives in Seattle booked her flight on the BA website, but her flight were on American both ways. You earn miles on American for your BA account as you would if you flew on BA.

Whoever is actually operating the flight is who you deal with.

If it's a BA aircraft making the journey--the airline operting--you deal with them and follow their rules. If it's AA, then it's them. It doesn't matter who you booked with or what country you are in.

Recent example of mine -

I booked my tickets entirely through Delta, BUT my flight home from London was on Virgin Atlantic. When checking in for my flight in London, I went to the Virgin Atlantic desk and obtained my boarding passes there.

Your flight may show up on the airport monitors with all of the flight numbers - AA - BA - etc. BUT if British Airways is operating the flight home from London (your ticket may say something like AA flight 1234 operated by British Airways) , then check in at the BA desk.