However,the meaning in the UK, I've been told by some Scottish girls who came here to live for a while, is similar (the same?) as cunt.

Similar, but they definitely aren't synonyms. 'Fanny' is a tee-hee, slightly juvenile sort of word, which (to get technical) implies the vulva/female genitalia fairly generally. 'Cunt' is much more ugly and explicit word for the vagina, more equatable with a particularly harsh inflection of something like 'gash'.