Gwynne's agent Alex Irwin said the actress died in a London hospital early on Friday, "surrounded by her beloved sons, close family and friends."
Born in 1957, Gwynne taught English in Italy before becoming an actor. She had her breakout role as a harried television news producer in 1990s sitcom Drop the Dead Donkey, a sharp but affectionate depiction of the world of journalism.
She also starred in medical drama Peak Practice and was a standout in royal roles, playing Queen Camilla in royal satire The Windsors and lady-in-waiting Susan Hussey in The Crown.
Her many stage roles included Prime Minister Margaret Thatcher opposite Helen Mirren as Queen Elizabeth II in The Audience.
Mirren paid tribute to Gwynne as "a delight as a person and a consummate dedicated actress."
"She was both funny and serious at the same time, a brilliant balancing act that her whole career exemplified. We will miss her very much," she said.
Gwynne was nominated for both an Olivier Award and a Tony for her performance as the supportive dance teacher in the musical Billy Elliot in London and New York, and received three other Olivier nominations for The Threepenny Opera, Women on the Verge of a Nervous Breakdown and City of Angels.
She performed in several plays this year, including The Great British Bake Off Musical, and was due to appear in the musical tribute Stephen Sondheim's Old Friends in London's West End. She withdrew from the production during rehearsals last month due to "sudden personal circumstances."
The show's producer, Cameron Mackintosh, said Gwynne was "a truly wonderful person, as well as a phenomenally talented actress and singer."Â
He said Friday's performance would be dedicated to her memory.
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