With a full head of thick dark hair, I think that adequate sunscreen on her part, as well as adequate sunscreen on her face would be enough. Even on a blistering hot day in England it's not the same as 30 degrees with blistering hot sun that you get being further south. I'm not saying that the sun does not burn in England or that you can be lax about it, but it's also a bit different than in more southern locations where you go outside and can be scorched within 10-15 minutes iyswim.
My DH is completely bald, my DS was pretty much bald until 18 months, and both wear appropriately and adequately applied sunscreen and we have no problems. (And summers are much hotter/sunnier here than England, regular days are 30deg, blue sky, no shade, sunny and hot!) Tops of ears, hairline, exposed parts of scalp, eyebrows, eyelids, etc. The hat helps shield eyes, all of those touch and tricky parts as well that we often miss with sunscreen but if you know she is not going to wear a hat then I think it is just a bit more diligent you have to be with the sunscreen application and some sunglasses for her eyes?