It opened sometime in the 1870s. The main landlords and landladies were Mrs Mitchell during the early 1900s then W. Stockwell took over the pub until around the time it closed in the late 1940s. W. Stockwell went on to run the Constant Service after this.
There is a lovely photograph of the Gardener's Arms on the Regency Society's Collection by James Gray:
http://regencysociety-jamesgray.com/volume27/source/jg_27_007.html