Wilkinson won a Bafta for 1997's The Full Monty, and reprised the role of Gerald when a Disney+ streaming series revisited the characters 26 years on.
He received six Bafta nominations in total as well as two Oscar nominations, for Michael Clayton and In The Bedroom.
He died suddenly at home with his wife and family, they said in a statement.
George Clooney, who starred with Wilkinson in the 2007 legal thriller Michael Clayton, praised the actor as "the epitome of elegance".
Paying tribute, he told Variety magazine: "Tom made every project better. Made every actor better.
"He was the epitome of elegance and he will be dearly missed by all of us."
With more than 130 film and TV credits in total, Wilkinson was as comfortable in period dramas like 1995's Sense and Sensibility and 2013's Belle, as he was playing criminal masterminds in movies like Rush Hour opposite Jackie Chan in 1998, or Guy Ritchie's RocknRolla in 2008.
He also earned an Emmy for playing US political figure Benjamin Franklin in 2008 mini-series John Adams and an Emmy nomination as John F Kennedy's father Joe in The Kennedys. He played President Lyndon B Johnson in 2014's Selma, and appeared in The Grand Budapest Hotel and Girl with a Pearl Earring.
His death on Saturday was confirmed in a statement shared by his agent on behalf of his family.
Actor Tom Wilkinson's memorable roles in pictures Wilkinson was, according to the Encyclopaedia of British Film, "a major character star, with a remarkable gift - one among several - for conveying inner pain".
Robert Carlyle, who starred alongside Wilkinson in The Full Monty, described him, as "a huge performer, a real titan of an actor, one of the greats of not only his, but of any generation."
In a tribute on X, formerly Twitter, British actor Phil Davis called him "powerful and delicate and hugely intelligent - one of the very best", while singer and actor Will Young described him as "British acting royalty".
US film maker Scott Derrickson said Wilkinson was "an amazing talent" and recalled a time when the actor gave him a surprise kiss on the lips on the set of The Exorcism of Emily Rose.
Born in Leeds before moving to Canada and then Cornwall in childhood, he found his calling at the age of 18 when he was asked to direct a play.
"For the first time in my life, I started doing something I knew how to do," he said.
"I realised it wasn't necessarily just these southern middle-class types that got to be actors; it could possibly be people like me. And once I knew, I never changed my mind."
He trained at the Royal Academy of Dramatic Art (Rada) before following the well-worn path to theatre and television work. In 1986, he got his first major screen role in mini-series First Among Equals, based on Jeffrey Archer's best-selling novel.
Playing alongside him was Diana Hardcastle. The couple married in 1988, and also went on to play husband and wife in The Kennedys in 2011, and in 2014 action film Good People. The couple had two daughters, Alice and Molly.
When he portrayed Pecksniff in Martin Chuzzlewit for the BBC in 1994, Wilkinson said: "I looked at it and I thought, I can't get it any better than that. It came out exactly as I meant it to come out. It won a couple of prizes and I thought, I can act, there's no question."
Three years later, he was cast as a former factory foreman who joins fellow unemployed workmates in staging a strip show in The Full Monty.
"I was simultaneously offered the lead in a TV series and a possible part in a low-budget movie," he told The Guardian.
"I remember phoning a friend and he said, 'Take the TV, take the TV'. But I didn't follow his advice, and the TV turned out to be crap."
The low-budget movie, meanwhile, turned out to be the highest-grossing British film up to that point, and took his career to a new level on both sides of the Atlantic.
Wilkinson received high critical acclaim for Todd Field's 2001 US domestic drama In the Bedroom, in which he played a bereaved father. He was nominated for an Oscar for best actor.
He hoped the film would do two things for him, he later said. "One, [prove] I could play the lead role in a movie. Two, I could play an American lead role. And it did both of those things."
The actor earned his second Academy Award nomination for his supporting role in Tony Gilroy's 2007 legal thriller Michael Clayton, starring George Clooney.
Wilkinson's other credits included Batman Begins, The Patriot, Eternal Sunshine of the Spotless Mind, The Gathering Storm, Black Knight, Valkyrie, The Lone Ranger and Denial.
He also provided the voice of the fox in the TV adaptation of children's best-seller The Gruffalo.
His long term agent Lou Coulson has described him in interviews as "one of the best".
Off screen, Wilkinson was known for being down-to-earth and keeping a relatively low profile. He said in an interview: "I like to go to Waitrose and not be recognised."
In the 2005 New Year Honours, Wilkinson was appointed Officer of the Order of the British Empire (OBE) for services to drama.
Source: BBC
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