Think it might be William Boxall Turpin from an 1868 article (which means the Globe WAS in existence!)
Morning Post 02 December 1868
The article is about law case where the prisoner seemed to be absconding with money from the local pubs, one being:
Mr William Boxall Turpin said he kept the Globe public-house, York-street, Bow-market.
In the 1891 census, William Walter Turpin was the manager of the Globe, at 8 York St.
In 1893, William Walter is listed (along with other Turpins) in a newspaper advert for claims against the estate of William Boxall Turpin, late of the Globe, but I don't think it was his father, perhaps an uncle.
Searching the history of the Globe, Boro' it shows on their website that York St. was changed to Bedale Street.
Edit:
South London Press 23 November 1867
List of businesses being fined for not measuring accurately:
William Boxall Turpin, York-street, Borough Market, the Globe: Six measures. Fined £1 10s
Edit again:
The Era 29 September 1861
From an article listing all the transfers:
St Saviour - Globe, York-street, Harriet James, widow and administratix of Henry James, the late occuper [sic], to William Boxall Turpin.
H. James (possibly the Henry James just mentioned) shows in an advert in 1827:
Morning Advertiser 24 August 1827
It's an ad about letting a shop, and at the end says For cards of address apply at Mr. Cooper's, Corn-dealer, No. 55, Union-street, Borough; or of H. James, at the Globe, Borough Market.
1827 is the earliest mention I can find of the Globe; I think whoever wrote up the history and put it's inception at 1872 either flipped the 27 into 72? It could be older, and just never made the papers, too.