Tye, just as there are regional American accents and dialects, the same thing is true in England. Also, in England the manner of speech is often governed by class and access to higher education. What you're probably thinking of as an "English accent" is the speech of someone born into an upper-class family and educated at Oxford or Cambridge. Contrast this with lower-class London speech, which is very harsh and clipped, or the speech of northern England (listen to old interviews with the Beatles and you'll get that), or Scotland or Wales.
I can't figure out what type of English Alizée is learning. As far as pronunciation is concerned, she definitely has a French accent. You can hear it in Fifty/Sixty, which she pronounces "Feefy Seexty." (It's terribly cute.
) So she doesn't sound either British or American. As far as word usage, at this point she's probably doing what I'm doing to learn French, and encountering a lot of English on the Internet (including here). God help her, poor thing . . .