OP's question uses the passive voice, asking whether people like her kids "are classified" as BAME, kind of implying that there's an objectively correct answer, or at least that there's, somewhere out there, an official BAME adjudicator whose job it'll be to pass judgement, but I don't really think there is here.
North African people whose recent ancestry is on the Med coast [most of them] can in most cases easily pass for someone whose ancestry is on the other side of the Med [e.g. Greece or wherever]. The offspring of someone like that & a white British person, well, they're clearly not B or A, I suppose they might, by some people including themselves, be classified as 'ME', but I'm not sure.
Geography does in a few cases throw up certain 'bright lines', e.g. if all of a person's ancestry going back 500 years is say Australian, or sub saharan African, they'll usually look a certain, easily recoginsable, way, but north africa isn't on any particular side of any such 'bright line', & a White British parent makes things even less clear-cut.