Cumberbatch played pioneering World War 2 codebreaker Alan Turing in The Imitation Game. Turing was prosecuted for being gay, but he was given a posthumous pardon in 2013.
The actor has signed an open letter calling for some 49,000 other men similarly convicted under the old law to have the same treatment as Turing. The petition has already been signed by almost 90,000 supporters.
The Duke and Duchess of Cambridge yesterday rejected Cumberbatch's plea to support the campaign, with a spokesperson saying that it is a matter for government, so they could make no comment.
Cumberbatch has teamed up with other public figures such as Stephen Fry for the campaign.
Baftas 2015 nominees
Baftas 2015 nominees
1/20 Baftas 2015 nominees
Rosamund Pike plays over-achieving Ivy League graduate Amy Dunne in Gone Girl and is nominated for leading actress
2/20 Baftas 2015 nominees
Michael Keaton (left) with Edward Norton in existential comedy Birdman - Keaton is up for leading actor
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Julianne Moore receives a devastating diagnosis as linguistics professor Alice Howland in Still Alice - she has been nominated for leading actress
4/20 Baftas 2015 nominees
Benedict Cumberbatch is up for leading actor for his role as code-breaker Alan Turing in The Imitation Game
5/20 Baftas 2015 nominees
Reese Witherspoon has earned a Baftas leading actress nod for her adventurous role in Wild
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Jake Gyllenhaal in plays a creepy journalist in Nightcrawler and is up leading actor
7/20 Baftas 2015 nominees
Amy Adams is nominated for leading actress for her role alongside Christoph Waltz in Tim Burton's Big Eyes
8/20 Baftas 2015 nominees
Eddie Redmayne is nominated for leading actor for his portrayal of physicist Stephen Hawking in The Theory of Everything
9/20 Baftas 2015 nominees
Felicity Jones plays Stephen Hawking's wife Jane Wilde in The Theory of Everything and is up for leading actress
10/20 Baftas 2015 nominees
Ralph Fiennes has a leading actor Baftas nod for Wes Anderson's The Grand Budapest Hotel
11/20 Baftas 2015 nominees
Patricia Arquette has a Bafta supporting actress nomination for her performance in Boyhood
12/20 Baftas 2015 nominees
J.K. Simmons stars as an abusive tutor alongside Miles Teller in Whiplash
Sony Pictures Classics
13/20 Baftas 2015 nominees
Keira Knightley is nominated for supporting actress for her role in The Imitation Game
Hand out press photograph/film still from the movie Imitation Game/downloaded from Papicselect
14/20 Baftas 2015 nominees
Ethan Hawke has a supporting actor nod for his performance in Richard Linklater's Boyhood.
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Emma Stone is up for supporting actress for playing Sam in Birdman
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Mark Ruffalo is up for supporting actor for his performance alongside Channing Tatum in wrestling drama Foxcatcher
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Rene Russo is nominated for supporting actress for her role as Nina Romina in Nightcrawler
Chuck Zlotnick/© 2014 Open Road Films/The Hollywood Archive
18/20 Baftas 2015 nominees
Michael Keaton and Edward Norton star in Alejandro González Iñárritu's Birdman - Norton is up for supporting actor
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Imelda Staunton (centre) with Liz White and Nia Gwynne in Pride - Staunton is up for supporting actress at the Baftas
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Steve Carell is nominated for supporting actor for his role as John du Pont in Foxcatcher
A letter to the government says: "The UK's homophobic laws made the lives of generations of gay and bisexual men intolerable.
"It is up to young leaders of today including the Duke and Duchess of Cambridge to acknowledge this mark on our history and not allow it to stand.
"We call upon Her Majesty's Government to begin a discussion about the possibility of a pardoning all the men, alive or deceased, who like Alan Turing, were convicted."
Cumberbatch has spoken passionately about the issue recently. "Alan Turing was not only prosecuted, but quite arguably persuaded to end his own life early, by a society who called him a criminal for simply seeking out the love he deserved, as all human beings do," Cumberbatch wrote to the Hollywood Reporter.
"Sixty years later, that same government claimed to ‘forgive’ him by pardoning him. I find this deplorable, because Turing’s actions did not warrant forgiveness — theirs did — and the 49,000 other prosecuted men deserve the same."
Stephen Fry has said he will launch a campaign to get Turing on the £10 note after the general election.