I’m afraid I know nothing about either the Free Gardenry or Sunflower Lodges but we’ll include your request in our forthcoming newsletter in the hope that someone may come up with something. Oxley is still very much a Hexham name; I remember Oxley’s tobacconist/sweetshop on Battle Hill well into the 1980s. In the period you mention, we have a listing for Henry Oxley, clog-maker, on Market Street in 1879 and another (?) Henry, blacksmith, on Eastgate between 1906 and 1935.
At the turn of the 19th/20th centuries, Hexham had several flourishing nursery gardens & orchards so it may well have had associated lodges
Holy Island is a group of houses at the bottom of Gilesgate, so named because it was bounded on three sides by burns, now culverted, and the largest house – dating from the 17th century, was known to have housed Catholic services. Quality Corner is more of a mystery!