The British Red Cross have records for an Arthur Badger, of 5 Stubbs Road, Wolverhampton, who served from April 1918 onwards doing various duties, including hospital work in the military wards of Wolverhampton General Hospital, the transportation of wounded at Wolverhampton and Lichfield, and air raid duty at Merridale Street Air Raid Station, Wolverhampton.
Tracking this man down has been more difficult, as there are various possibilities for who it could be. The 1911 census shows an Arthur Badger, born about 1872 (there are in fact, the births of two Arthur Badgers registered in Wolverhampton in 1871), living at 28 Rayleigh Road, Wolverhampton. This appears to be the most likely man, as he is the only one living in Wolverhampton in 1911, but there are others with the same name living in the surrounding area.
This particular man is married to Clara Helen Badger, and the have a 12-year-old daughter, Gladys. Arthur is a coal merchant. His age in 1911, 39, again makes it likely that this is the man in question, as he was probably too old for conscription by the time of the First World War. This particular man died in Wolverhampton in 1948, at the age of 77. If anybody is able to provide any further details, or confirm that this is the right man, we would love to know!