I have my new bank card. So I call up the DVLA to order a replacement driving license. Turns out that I can't order over the phone if I've only lost one part of my license - I have to return the counterpart through the post, and hence I have to fill in the form, write a check, post it, etc. Much hassle. I hang up.
They didn't ask my name. I checked that there isn't a price difference between losing one or both parts of the license - there isn't. So I just called straight back, told them I'd lost both parts, and they let me order it then and there with my card.