If you are already a UK citizen and have a digital UK passport you can just enter the passport number on your application form for a provisional driving licence as id. You don't even have to send a photo, DVLA can use the one from your passport. You don't have to send a passport at all.
But if you don't yet have a UK passport this can't be done. You have to send your US passport to the DVLA as proof of identity. So it can be a worrying time waiting for it to come back - one of your most precious possessions for so many reasons.
My partner sent the application special delivery enclosing passport and application and a prepaid special delivery envelope self-addressed for the return of the passport. As per one of the previous posters. Note the tracking numbers of both envelopes so you can check on their progress. The whole process took only a week. Sent Monday, passport returned Saturday and licence received Monday. But me - worried sick ....