I'm sorry, it was me who wasn't clear! Actually, it's rather odd, because I think most Brits don't think of a generic English accent, but a collection of different English accents (Scouse, Geordie, Mackem, Cockney, etc.). If I said somebody didn't have an English accent it could mean they could have any other accent (whether or not the accent was from an English-speaking country, so all of your possibilities above - Scottish, French, etc.). But it could also mean no distinguishable accent.
I agree that he refers to both (language/nationality, and accent) in the same sentence! Meaning you sound English (accent), but you aren't English (nationality).
(I feel that he's implying that she's faking an "English" (i.e. from England) accent, rather than that she just happened to pick one up while she was here. But I could be wrong).