Upvote:2
This is probably not the best solution, but if the US does not approve the ESTA application with your UK passport then the only option other than changing your flight will be to go to a US consulate and apply for a new passport. They can issue an emergency passport very quickly, though I believe this will result in the cancellation of your existing passport. The US does issue second passports to travelers with a demonstrated need for one, but I believe it requires submitting your existing passport with the application, and I suspect that consular officers generally won't consider failing to anticipate a change in plans to be a genuine need. Still, it can't hurt to ask.
Upvote:3
Tricky. As a US citizen you are required to use your US passport to enter the US but there is no penalty for not doing so and you can't be denied entry. However you need to somehow proof to the immigration agent that you do have a US passport and they can drag the whole process out as long as they feel like which clearly includes you missing your connection.
If you can get a photocopy of you passport and birth certificate or naturalization certificate and what ever other proof of US citizenship you can muster you may get lucky: approach the immigration officer with the full story: why you don't have your passport on you, that you only be in the country for a few hours, that you have proof of imminent departure and that you try to collect as many documents supporting your story as was feasible.
A "reasonable" immigration officer will let this go, but there is, unfortunately, no guarantee for that. A grumpy one may look at your boarding pass for the connecting flight, and keep you there just long enough to miss your connection.
I would also contact the airline, explain your situation, ask for advice and specifically ask what would happen of you miss your connection because of immigration delays.