I have a bit of information concerning your gg grandfather George Cole Bainbridge.
I have been researching my Hodgson ancestry in Jamaica for nearly 4 years now and have collected many documents concerning people with that surname if they lived or had connections in Jamaica.
George Cole Bainbridge, Merchant of Liverpool, appears as an executor to a will of 'Wine' Hodgson of Saint Mary and Saint Ann, Jamaica, this will was proven at London 5th February 1828. Wine lived on a Plantation and had slaves in Jamaica but died in Wem, Shropshire. The other executor Richard Walford was of Wem also, she gave them each £30. One of her Hodgson grandson's was an 'absentee' landowner of Jamaica prefering to live in the Liverpool and Shropshire area's
Wine was born 1758, possibly in England and seems to be of an Hodgson family that had links to Huddersfield or Halifax in Yorkshire, I think I spotted a Saint Peter's Church in Huddersfield or Leeds some time back, but I'm not 100% sure. She was a daughter or relative of William Hodgson of Saint Ann, Jamaica. He died in America 1793, one of the signed witnesses to his will was a Thomas Bainbridge, I would guess he is related to George Cole
On the Public Records Office website, for £3 you can download the will of a 'Thomas Bainbridge of Saint Ann Jamaica'
Regarding Jane Hobson? maybe this was a mis-spelling and she was infact a Hodgson! There are a few Jane Hodgson's in their family.
I dont know if these Hodgson's were related to my Jamaican Hodgson's. Let me know how you get on or if I can help further.