Final approval of the anti-trust exemption to allow BA and AA to codeshare trans-atlantic is AT LEAST 60 days away (though all-but-inevitable now with the preliminary approval from the US DOT), and codeshares won't necessarily start immediately even then. Even when codeshares do begin, you will always know at booking if it your flight operated by an airline other than the marketing carrier. For example, the booking information will say "BA 7654 operated by American Airlines".

The anti-trust immunity and expanded codesharing are not a merger, so the two carriers will retain their separate identity; they'll be sharing revenue and coordinating schedules on trans-atlatic flights.