If you were born in England with some German ancestry that wouldn't rule you out of being British.
Unless they had a German surname, someone with a German grandparent in England would be almost unidentifiable unless they told you.
There isn't any reason to feel any differently to people born outside of England or Britain.
Genetics is only one part of it anyway. Asian immigrants born in England can be considered British (if they have assimilated into our culture).
Mmm...interesting point. So how much would the metrics culture and lineage be weighted when creating the mix: British. Though is American culture close enough to British culture that Anglo-Americans (by ancestry) can be called British, since they all ready qualify by parent hood. Or have I simplified it far to much.