Although born in India, Uzzaman moved with his family to Pakistan as a child. He became a radio announcer and acted in Pakistani television. However, after appearing in the 1982 film Maila (in multiple roles), the movie's political content ran afoul of the then-military government. Uzzaman went to Great Britain as a political refugee, gaining asylum and continuing his career, usually as an assortment of shopkeepers or vendors.
After a stint on BBC Radio (including various roles in the 1984 serial Masters' India), Uzzaman appeared in the acclaimed film My Beautiful Laundrette as a drug dealer. He had a larger role than usual in Brothers in Trouble (1995), about illegal Pakistani immigrants. Other film credits include Cry Freedom (with Denzel Washington and Kevin Kline), Sammy and Rosie Get Laid (with Shashi Kapoor), Bellman and True, Son of the Pink Panther, and The Fourth Angel (with Jeremy Irons and Forest Whitaker).
On television, Uzzaman appeared in the first installment of the miniseries The Singing Detective (1986), as Michael Gambon's hospital bed neighbor Ali. He was a regular on Tandoori Nights (1985-1987) as Rasheed and recurred on Specials (1991, as matchmaker Uncle Ram). He also appeared on the Jeremy Brett Sherlock Holmes series (in "The Sign of Four"), EastEnders (a brief stint as Hanif, 1990), Inspector Morse, The Bill (multiple episodes), A Touch of Frost, Torchwood, Prime Suspect (with Helen Mirren), Hetty Wainthropp Investigates, and Casualty. In the TNT miniseries The Grid, he appeared in most installments as a hotel desk clerk.