It is strongly implied by his uncle, Arkwright, that his mother (Arkwright's sister) was an unmarried mother who died young, leaving young Granville to be brought up by his uncle.
Assuming that his mother registered Granville's birth, should would almost certainly have done so under her maiden name which would, obviously, also have been Arkwright.
In which case, I think it can be safely assumed that Granville's surname must be Arkwright too. Can anyone think of any sensible reason why it wouldn't...?