Agnew was born as Mona Christine Nimmo Duggan in Altrincham, then Cheshire (now Greater Manchester), England on 29 December 1899. She was youngest of four children born to Dr. Motherwell Duggan Jr. and Wilhelmina Nimmo, who married in 1883. She had three oldest siblings: Edwin, Norman, and Eva. Her paternal grandparents were Irish.
In 1923, Agnew married solicitor Norris Montgomerie Agnew. The couple had three children. Their eldest child, Joan, died as a newborn in 1924. They later had Margaret in 1926, and Philip in 1929. One of her grandsons, Jonathan Agnew, was a professional cricketer and later a cricket broadcaster.
Agnew’s husband was awarded a CBE in 1957, and was knighted in 1967. He died in 1973 after 50 years of marriage.
Agnew died at Priory Court, a nursing home in Stamford, Lincolnshire, England on 17 June 2010 at the age of 110 years, 170 days.