Massey came to Britain with her parents in 1946 at the age of 12. Her father fought in the Second World War, but remained in the UK and became a Producer for the BBC.
She began her career as an extra. Her first film role was in Sink the Bismarck! (1960), where she was used to translate and report a news section from the film into Urdu — announcing the sinking of the British battleship HMS Hood.
She continued to work extensively on radio in English, Urdu, Hindi and Punjabi as well as for the German Service of the BBC. Years later she acted in Tribunal, a ZDF television serial in German, with leading European actors. She joined the Afro-Asian Committee of Equity and continued to work in film and television. Roles for Asian actors were scarce during her early career, but she was given parts in series such as Crossroads (1964); Within These Walls (1976); The Next Man (1976); Z-Cars (1976–1977); Target (1977); Mind Your Language (1977–79); Play for Today (1977, 1978); Empire Road (1979); Angels (1981); Minder (1982); The Jewel in the Crown (1984); The Bill (1984); Albion Market (1985); Chance in a Million (1986); Madame Sousatzka (1988); Great Balls of Fire! (1989); Brookside (1989) playing Manju Batra and Pie in the Sky (1994).
In the mid-1990s she was cast as the recurring character Auntie Satya in Radio 4's daily agricultural soap opera, The Archers, fulfilling one of her life's ambitions. Massey was the second Actress to play the role. She had initially been asked to audition for the part, but was forced to turn it down due to work commitments. A year later she was offered the same part when the role was recast. Her character makes occasional appearances to visit her niece, Usha Gupta (Souad Faress).
In 1997 she was cast in the popular BBC soap opera EastEnders. She played Neelam Kapoor, the domineering mother of market-trader Sanjay (Deepak Verma) until 1998.
Other acting credits include Arabian Nights (2000); Adrian Mole: The Cappuccino Years (2001); Perfect World (2001); Doctors (2002); All About Me (2003); Chicken Tikka Masala (2005).
Massey appeared in Coronation Street in February 2010 as one of Sunita's aunts.