We have to keep in mind that there is no recording or transcript, and the conversation as reported is one person's memory of it some time later. The BBC is irresponsible for presenting the exchange as exact dialogue, without a disclaimer - it certainly looks like a transcript, rather than an approximate recollection. I believe it's directionally accurate but am not taking it as exact. I agree Hussey went too far: Fulani was uncomfortable, and the conversation should have stopped at that point.
That said, intentions matter. I think it's just as likely that Hussey was trying to celebrate diversity as she was being racist. Someone dressed in African garb representing an Afro-Caribbean charity probably has a personal story / connection, and I think that's what Lady Hussey was trying to draw out (however awkwardly). People in the US, where I was born and where my accent is standard, ask me where my family is from all the time simply based on surname. Replace Africa with Italy, and replace London with Philadelphia, and I've had this conversation countless times despite being second/third generation. My grandparents had it even more frequently. I think asking once is OK. Our histories are a part of us. Of course, anyone can choose not to discuss, and that choice should be respected.
I do think "where are you from" is a loaded question that's better avoided. In this globalized world, people are "from" all sorts of places, and as I said, I often feel like I'm from nowhere. Where do you live, where did you grow up, do you have a personal connection to Africa / Italy / whatever, all fine in my book. With a little bit of tweaking, this conversation could have easily not gone off the rails.