In my youth, a "British Accent" was that used by BBC announcers - anything else was a regional accent. But I don't think one could argue that case any more, as you say, Jo. I was going to say that it doesn't matter so much in the UK, and that it is only people "abroad" who might wonder about a "British" accent. But in fact, the more multi-cultural this island becomes, the more one is aware of "foreign" accents being used.
For me, the question is, When does somebodies accent, which they gained in their country of origin, or from parents from off-island, become a "British Regional" accent? For instance, I don't think of the accent of people from the Indian sub-continent or the Carribean as "foreign" but as "regional" (or perhaps as "cultural") But mt Jugoslavian neighbours, who had lived in England for nearly thirty years, definitely had a "foreign" accent.