Is it possible to find out the BVA hip score of a dog who isn't KC registered, and is a rare breed?

By asking the owner?

Or the breeder if the owner doesn't know. Should be able to show proof too

No, not unless there is some sort of pedigree database and the owner as inputted it in there (although this could be inaccurate unless the site requires proof). You would have to directly ask the owner.

There is the English shepherd club registry. They say health info is one of the things they make advaible, are UK breeders using this site?