The following is what he needs:
- Check-in for flight to the US: US passport
- US border control: US passport (at most airports he'll use a kiosk to scan his passport and get a receipt which he'll then present to the officer)
- Check-in for flight to the UK: British passport
- UK border control: British passport (at most airports he can use an e-passport gate and won't need to see an officer at all)
Absolutely no other paperwork is needed for the UK - again he almost certainly won't need to see an officer to begin with.
For the US, TIMATIC, the database used by airlines, says:
Minors traveling unaccompanied, or accompanied by one parent
or a person other than parent/legal guardian, can prevent
possible delays if holding a notarized letter of
authorization signed by absent parent(s). For more details
refer to
https://help.cbp.gov/app/answers/detail/a_id/3643/kw .