If you became a US citizen, but have not yet got an American passport, can you fly from the US to the UK using your British passport?
You can (and should) fly from the US to the UK with your UK passport. Technically, you should have your US passport with you to depart as well, though this isn't enforced.
The problem is that you can't fly back to the US with a UK passport; you must enter the US with a US passport. If you need to travel soon, you can visit a passport agency to get one in a hurry. It's also possible to get a US passport outside the US, but it may take longer and you might not have all the required documents with you, so it's best to get your US passport before you leave.