There appears to be no connection between the South Tawton Oxenham family and any other branch of the Oxenhams. I have a copy of a family tree produced about 1800 to prove that the last surviving male member was William Long Oxenham, a lunatic, who died in Sidmouth in 1813. His sister Elizabeth married into the Ackland family and inherited the Oxenham estates in South Tawton which I think is the reason the family tree was created to prove that she was the legitimate heir.
Little is known about Capt John Oxenham although his voyages are well documented. He is likely to have come from Plymouth and there were certainly Oxenhams there in the mid-16th century as there are Oxenham Plymouth wills of that period in existence.