should not be fully reassured until at least 3 months have passed during which they remain sero-negative (the window period).
The network of public clinics in London is first class.
If you turn up early and are prepared to wait they will see you, though they prefer an appointment to be made.
At some clinics eg Lloyd at Guy's & St Thomas's you need an appointment for a rapid test which are only available certain days . The Doctors prefer to test by DUO anyway -it is cheaper, they have more experience with them as rapid testing is only in evaluation.
If you want details of GUM clinics and THT FASTest clinics check check the THT website
GUM:
http://www.tht.org.uk/howwecanhelpyou/hivandstitesting/gumsandhivclinics/
FasTEST
http://www.tht.org.uk/howwecanhelpyou/hivandstitesting/fastest/