Grima was born in Rome to Italian-Maltese parents and raised in London, where he attended St Joseph's College, Upper Norwood. He later studied mechanical engineering at Nottingham University. The family was artistic and creative: his father was an embroidery designer and Grima's brothers became architects, helping design his London showroom in 1966.
Grima joined the Royal Engineers, serving in Burma in World War II with the 7th Indian Division of the British Army.
After the war, he started work in his father-in-law's jewellery firm in London, initially in the accounts department, before moving into design.
He became the foremost modern jewellery designer in the West End of London in the 1960s and 1970s, selling designs from his exclusive gallery in Jermyn Street, Mayfair featuring the world's first perspex spiral staircase (1970) built by Peter Rice and Ove Arup. In 1970 he designed the successful "About Time" watch collection for Omega and in 1976 a collection of gold digital LED watches for Pulsar
Grima won several awards for his contribution to the jewellery industry. Namely, he was the only jeweller to win the Duke of Edinburgh Prize for Elegant Design and won 13 De Beers Diamonds International Awards; more than any other jeweller. His clients included members of the British royal family, as well as Barbara Hepworth and Jacqueline Onassis.
Notable pieces include Queen Elizabeth II's ruby brooch, as well as a brooch of lichen cast in gold for Princess Margaret. More recently, fans of his work have included fashion designers Miuccia Prada and Marc Jacobs.
Examples of Grima's work are held in the Victoria & Albert Museum and in the collection of the Worshipful Company of Goldsmiths.
Grima married twice, firstly in 1947 to Helène Haller, niece of the Viennese jeweller who brought Grima into the trade. The couple had a son and two daughters; they divorced in 1977.
The same year, Grima married Jojo Maughan-Brown, great-granddaughter of Sir Thomas Cullinan. They had one daughter, Francesca, who – together with her mother – continues the family business.
In 1986 the Grimas moved to Switzerland, first to Lugano and in 1992 to Gstaad, where Grima died on 26 December 2007.
Francesca and Jojo moved back to London in 2012. They have continued the family business by creating unique and bespoke pieces handmade in Britain by Andrew's master craftsmen.