How will the UK ever know you've left the country though?
I believe they do exit checks now.
The US passport must be used to leave the UK because:
a) the airline will check that everyone on the flight holds either a US passport or a valid US visa/ESTA
and
b) as a US citizen, it is illegal for your daughter to enter the US on a UK passport... so she has no choice - she MUST use her US passport for the flight booking, for checking in and for entering the US
Basically:
- by law, all US citizens must travel to and from the US on a US passport
- there is no such law for UK citizens, so she can use either passport to leave, and then she needs to use her UK passport to return, so that they know she is a UK citizen
So, in summary, she uses her US passport for the majority of the trip (leaving the UK, entering the US, leaving the US) and then just shows her UK passport to UK immigration when she returns.
