Nicholas is the son of Bridget (née McCall) and Harry Oppenheimer, and grandson of Ernest Oppenheimer, founder of Anglo American and the first generation of the family to chair (from 1929) the De Beers Diamond mining company in South Africa, established by Cecil John Rhodes in 1888. His Father was of German Jewish descent. Oppenheimer was educated at Harrow School and Christ Church, Oxford, where he read Philosophy, Politics and Economics and took the Oxford MA.

In 1968, he married Orcillia "Strilli" Lasch, daughter of industrial tycoon Helli Lasch; both are Anglican. His Father was born Jewish and converted to Anglicanism.

Oppenheimer was appointed Deputy Chairman of the then Central Selling Organisation (now Diamond Trading Company) in 1984 and Deputy Chairman of De Beers Consolidated Mines in 1985. He was appointed Chairman of the Diamond Trading Company in 1985. He was Chairman of the De Beers Group from 1998 to 2012, at which point he retired when the family stake was sold to Anglo American.

In 2003, the Technikon Witwatersrand in South Africa bestowed Nicky Oppenheimer with a Doctorate in Technology, honoris causa. He was the first recipient of such an honorary doctorate by the Technikon to a person in the public and private sector. He has also received the Presidential Order of Honor (2004) from the former President of the Republic of Botswana, Mr. Festus Mogae, and an Honorary Fellowship (2009) from the London Business School.

The Oppenheimer family has directed much of its philanthropic efforts towards preserving the heritage and cultural importance of the Southern African region, as well as to broader community upliftment in the areas of education, health, nature conservation and the arts. Nicky Oppenheimer and his son Jonathan Oppenheimer established the Brenthurst Foundation in 2005 as a way to contribute to the debate around strategies and policies for strengthening Africa's economic performance and enabling inclusive and sustainable development. The family has also long been involved in environmental and conservation issues. The Oppenheimer family partnered with De Beers to establish the Diamond Route in 2006 to maximise the potential of their properties for conservation, research and environmental awareness purposes. The Diamond Route links 8 sites across northern South Africa, stretching from Namaqualand on the west coast, to Kimberley, north to Tswalu in the Kalahari, and to the Brenthurst Gardens in Johannesburg, eastwards to Ezemvelo Nature Reserve and northwards to the Venetia Limpopo Nature Reserve in Limpopo Province.