Swiss-Italian family. Restaurateurs, music hall, theatre and electricity supply entrepreneurs. Four generations of note, all born in Dongio, Val di Blenio, Ticino, Switzerland: Carlo; Agostino (1841–1897) and Stefano (1844–1906) who were sons of Carlo's brother Giovanni; Sir John who was Agostino's son; and Jack (1898–1972), Sir John's son.
The ODNB has a full entry for the family.
2021: The Southern Daily Echo (supported by Encyclopedia Titanica) tells the story of another London Gatti; Gaspare Antonio Pietro Gatti, known as Luigi, born 3 January 1875 in Montalto Pavese, Italy. He came to London as a young man and set up restaurants including "Gatti’s Adelphi and Gatti’s Strand". It's difficult to reconcile this Echo story with that told by the ODNB which has these restaurants set up by Agostino and Stefano Gatti. Someone's got their Gattis in a twist. We can't untangle them, nor discover how they were related (which surely they were). Where's an Italian genealogy consultant when you need one?
According to the Echo, Luigi also managed the restaurant at the Ritz and ran the Ritz concession restaurants on ships, first the Olympic and then the Titanic, which gives you his date of death: 15 April 1912.