I had a leg of my recent northern Italy trip on BA, that being on the first leg of the return flight from Malpensa to Heathrow, where I caught an American flight to DFW. The transatlantic flights both ways were premium economy. I've flown business class transatlantic once, and I doubt I would do that again as I just can't sleep on a plane, even in lie flat seating (however, lounge access was nice in ATL when the airport shut down briefly due to a thunderstorm that parked over the field).
When you book one of these flights through American, you can't choose your seat on the BA flight unless you do it separately through BA, which requires you to jump through a bunch of hoops and pay $59. I did this right after booking my flight in July, and chose a window seat, 11A.
When I checked in for my flight at 7am on Saturday morning (flight out was 7 am Sunday), BA had changed my seat assignment to 12D. I wasn't pleased with this (I'm one of those folks who just enjoys looking out the window, and don't like getting brushed by everyone going up and down the aisle), so I tried to get that changed, both online and at the gate, to no avail. They said, "Perhaps the aircraft was changed and has a different seating configuration," to which I replied, "The new plane still has window seats." I was told I could request a refund, which they MIGHT grant in 28 days, so they were intending to hold the money that I paid for selecting a seat for about a year.
Even though I paid to pick my seat and was towards the front of the plane, I was in the last group to board (5), and there were only 15 or so people behind me in line. When I got to 12D, I noticed that 11A and 11B were empty, and, after a few minutes, aske the flight attendant if they had completed boarding, and, if they had, whether I could take the seat I had selected and paid for. She told me that she would let me know when boarding was complete, but if the seats were unoccupied, I could take 11A. A few minutes later, she came back and told me I could take the seat. So, not only did I end up with my seat, but I think the middle seat was the only unoccupied seat on the entire flight.
I'm convinced that there was never really anyone that was going to sit in 11A, it was just some computer glitch. And, I was surprised when I saw that "first class" on that flight was exactly the same seating as the seats in economy (at least on the exit rows, where I was seated--maybe it was tighter in the back).
Combine this with the fact that if you are going from BA to American at LHR, you have to clear security again in the UK, and it's just a really unappetizing prospect. In the future, I'm going to do everything I can to avoid flying BA.