At least one species of Queen Anne's Lace is not just called wild carrot, it is literally the same species (Daucus carota) as domesticated carrot. The roots are said to be much woodier in texture than domesticated carrot, though. This might be exacerbated by the fact that Daucus carota is a biennial, and the carrots we eat are harvested during their first year when they are more tender. By the time Queen Anne's lace flowers, it's already in its second year. If you can identify any first year Queen Anne's lace, the roots might be worth a try - but anything with those easily recognizable flowers will have woody roots.
(If you garden, domesticated carrot
should happily grow anywhere that Queen Anne's lace grows, but because they're the same species cross-pollination makes seed-saving chancy.)