I have US/UK citizenship. They tell you that you have to use your US passport at all times, which I have done Maybe it is best to stick to your US passport and just suck it up if the immigration lines are too long (I'm looking at you CDG).
When I checked in using the iPhone app it asked me which documents I was using. I added my US passport and had to take photos using the app. As I recall it was a bit hit and miss. I might have had to show my new passport at the counter , I can't remember.
If you have Global Entry on a green card attached to your UK passport then that is a bit of a pain. I showed up at the downtown NYC processing center and wasn't allowed into the building without an appointment. I tried to make an appointment but found it was only for new applications. I tried to change my documents online (new passport) but that only works for US->US passport changes. Eventually I went to EWR a few hours before my next flight as they take drop ins. After 35 minutes of hanging around in front of a sign saying "do not approach this door" and being roundly ignored by everyone inside the glass walled office I approached the door, got yelled at, but eventually got processed (after a lecture on security).