Glad this topic was resurrected, it prompted me to try again to find the Hill grandparents of Caroline Hannah Hill. This time I believe I might have found them, a Daniel and Sarah Hill who had five children baptised 1783 to 1791 in Beddington Surrey, one of them a Daniel Hill born 1788 who might be the father of Caroline.
Unfortunately it wont be an easy task to link these possible grandparent, Daniel and Sarah Hill to Caroline's parents Daniel Hill and his wife Charlotte Louisa Russell because of the five children the possible grandparents, Daniel and Sarah, baptised, only the one named, as you would expect, Daniel, matches any names used by Daniel and Charlotte Hill. Although Caroline's parents Daniel and Charlotte did name a daughter Sear, not sure if that is a variant of Sarah.
When Caroline's parents, Daniel Hill and Charlotte Russell, married at St George Hanover Square, Daniel was stated as being of Wandsworth, Wandsworth looks to be around 8 miles from Beddington Surrey.