I'm not sure if you are looking for Armistead Beresford Cole or not. His lifespan was 1822-1873 and his wife's name was Sarah Jane Turner.(1824-1876). He was born in Alabama and died in Texas. His ancestry in England has been traced back to the 13th Century.
He was not a descendant of Emilia Cole-Hamilton. His ancestors lived in the US and the colonies before that after they came from England. I don't know how he wound up with a middle name of Beresford. So perhaps, his Cole family had connections at one time with the Beresford's.