Carrots are remarkably sweet -- about 50% sugars by dry volume. The UK probably didn't ever need to refine sugar from them given that sugar beets also grow well there, but I wonder if they could have done so in a pinch.
My grandparents made dandelion wine and potato wine. My dad tried making it a few years back for a trip down memory lane, and they really were not the sort of thing that you would want to drink much of- given how cheap everything is nowadays, it's more for the novelty.