Upvote:0
You shouldn't have any issues. In fact, US passport holders can use the automatic eGates in the UK, so it is unlikely you'll have to speak to an officer at all, unless the gates are off or not working properly. Just place your US passport on the scanner, look at the camera, and wait for the gate to open. Do bring your expired UK passport anyway, though, to present with your US passport if you have to be processed manually.
Upvote:7
According to https://www.traveldoc.aero, an expired UK passport is sufficient by itself to fly from Switzerland to the UK if the passport has been expired for less than ten years.
You'll need to show the US passport to the Swiss border officers (or other Schengen officers, if that's the case) at the at the departure passport inspection, because you will have entered with that passport.
It's probably also a good idea to show the US passport to the airline to avoid complications. If they ask about your duration of stay in the UK, you can show the British passport, but they probably won't ask.
There's no need to show the US passport to the UK immigration officer on arrival. There's also no harm in showing it, so do whatever seems likely to result in the most efficient experience.