I asked about him on the Great War Forum, and SteveE came up with two possibilities - "There are two J. McGee(s) listed on the 1914/1915 Star Medal Roll for the Northern Rhodesia Rifles. #19 Private J. McGee and #75 Private J. McGee (deceased), both recorded as entering theatre of war on 2nd February 1915. Based on the article if it’s either of the men listed then the former looks most likely." and "#19 was Rifleman John McGee (#75 was Jack McGee) of the North Rhodesia Rifles. The British War & Victory Medal Roll (NA ref. WO 329/2336) has them listed as “Magee” and John served in the African Theatre (a) N. Rhodesian border from 2/2/1915 to 15/11/1915 so there is some evidence at least that the story has some basis in fact."
I had a quick look on the relevant rolls. There is one "Tpr. J.H.McGee" who is on the Owen roll for the Rhodesian Horse Volunteers and earned the 1896 medal with 1897 clasp. Regarding his being "with B-P when Mafeking was relieved", there is no entry in Colin Walker's Mafeking Siege Register. However, there is one "n/n Tpr. McGee, J" on the Kaplan roll of the Relief of Mafeking clasp for E Squadron of the Rhodesia Regiment. Clasp entitlement was RofM, Rhod., OFS and Transvaal. Running the thread along, the Kaplan roll says that the same chap also served with C-inC BG as 22454 (clasp SA1901" and also was an FID Scout (Buxton roll); earning the SA1902 clasp.
Well, if it the same chap, that certainly is the traditional "chestful of medals".