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Johnston was born in San José de Mayo, Uruguay. His father, Fowell Buxton Johnston (born 1839), was an officer in the 3rd Dragoon Guards, and the younger son of Scottish MP Andrew Johnston and his second wife, abolitionist Priscilla Buxton, daughter of Sir Thomas Fowell Buxton, 1st Baronet. Johnston's uncle (his father's elder brother), also Andrew Johnston, became an MP in Essex in the 1860s.