Yes, you can fly privately here on your FAA certificate.
Any flying school from which you wish to rent would require you to do a check ride with an instructor, just as in the States.
Beyond that, additional requirements/conditions will vary between schools so it's worth shopping around.
I refer to schools because, unless you know someone with whom you come to a private arrangement, virtually all self fly hire of light aircraft in the UK is from flying schools.
It would be wise to bring your logbook (or a photocopy of recent years' entries) to show your hours, experience on types and currency.
Heliport