Hi JP,
Your Harts need to go back at least one more generation before we might find a link.
My line of ancestors and their siblings go from an ancestor John Hart born circa 1665 to an ancestor's sibling baptised in 1809. The line ends with Sarah Hart marrying William Olley in 1813, the Olleys then married into the Theobald line.
My Hart known locations are:
1627 to 1668 Great Waltham, Essex.
1689 to 1758 Panfield, Essex.
1788 to 1800 Braintree, Essex.
Please contact me if you get back to a time or place where we can compare notes.
Good luck.