I would add onto this to clarify what I have seen others speak about when travelling between US<->UK as a dual citizen of both countries (this may apply to other countries).
As a US citizen you must exit / enter using your US passport; so when you arrive at the US border control you hand your US passport over.
As a British Citizen, when you arrive at the UK border, you should present your UK passport, so that you are admitted in as a UK citizen with no requirement for an ETA (electronic travel authorisation) and no restriction on your stay there, even if it is only temporary.
You may need to ensure the Advance Passenger Information System (APIS) data is updated correctly before checking into your flights, i.e ensure that on the return flight to the US it is your US passport info that is entered, and on the outgoing flight to the UK you may need to ensure it is the UK passport info so that the airline do not assume you need a (UK) ETA.