The only thing now is that some airlines such as United and American have switched over to mandatory self check-in. You check-in either at home or at a computer kiosk at the airport and then drop off your bags if you have any at a counter staffed by non-airline personnel.
My experience has been mixed. I don't have a British passport yet, but it is sort of similar because I use my US passport to enter the US, and I use my Canadian passport with ILR stamp to leave the US/enter the UK. Just like you would with a British passport
With American Airlines it has been fine. If you enter the US on a US passport, their computer system has no record of your other passport. So when you return to the UK, that information needs to be entered. Fortunately in my experience, there is always an American airlines manager-type hovering around the check-in kiosks at the US airport who can verify your second passport or do a visa check.
United Airlines was not a good experience. The self check-in area at LAX was chaos. I tried to check-in with my second (Canadian) passport, but the system wouldn't let me. I had to see a United employee, but the baggage drop-off was staffed by airport employees, not United ones. There was just one United employee serving over 50 self check-in terminals, and it took a very long time before he finally got to me to enter my second passport information on the system so that I could legally (in their eyes) enter the UK.
I have never had a problem travelling to the US showing just my US passport both in London and in the US. The problem has often been coming back, when people on the US side want to see the second passport to enter the data. Or when someone at the boarding gate wants to get my green I-94 visa waiver stub before allowing me on the plane, which as a US citizen I don't have. Dual (or multiple) citizens like us need to make sure we carry both our passports at all times to prove legal entry to and legal departure from both the US and the UK. It's just departing from the US where the procedure gets a bit fraught. But again, your mileage may vary.