Benedict Cumberbatch is concerned that the world might be growing tired of him. Though the actor, who shot to fame as the titular character on the BBC series “Sherlock,” has long had a rabid and vocal fanbase, he is rapidly emerging as a leading man. His star turn in last year’s “The Fifth Estate” badly stumbled, but he more than compensated with wildly different roles in “August: Osage County” and “12 Years a Slave.” And now he’s tipped for his first Oscar nomination, playing World War II codebreaker Alan Turing in the just-released drama “The Imitation Game.”

Cumberbatch, making a brief stop in Los Angeles from the U.K. set of BBC miniseries “The Hollow Crown: Richard III” to do some press for “Imitation Game,” is enjoying the attention, despite his uneasiness about being overexposed. “The more work you do, the more publicity you have to do,” he says. “That’s the only time I get worried — the idea that people might get sick of me not because of what I’m doing as an actor, but because of the proliferation of me in the media.”

In an interview with Variety, the whip-smart 38-year-old British actor is often self-effacing and witty. On being in bed in England when he found out he had won the Emmy Award for “Sherlock” in September: “I literally won in my sleep,” he quips. His biggest regret was not being able to make the ceremony and receive the prize presented by “True Detective” co-stars Matthew McConaughey and Woody Harrelson. “They announced my name and just wandered off. Probably smelted the statue into a bong by now.” And while he’s honored by the win, he jokes that it comes with “a ludicrous subtitle. Something like, ‘Lead actor in a motion series or drama single episode with Martin Freeman in it but not the one with Moriarty.’”

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What he is deadly serious about is his work in “The Imitation Game.” When Cumberbatch first read Graham Moore’s Black List script, it was set up at Warners with names like Leonardo DiCaprio and James McAvoy being talked about to play Turing. “I was on a general meeting at Warner Bros., and didn’t know who was sitting on it when I mentioned my interest,” he recalls. “I found out Leo was on it, and thought, ‘Well, I’ve wasted a trip.’” But for the first time in his career, says Cumberbatch, he tracked a project. “It wasn’t until a year later that it really crystallized for me.”

DiCaprio fell out, and with him, Warner Bros., and when director Morten Tyldum signed on, it was with a different plan. “Benedict was my first and only choice for Alan,” Tyldum says. “When you think of an actor who can convey genius and vulnerability, that’s Benedict.” It was a role he didn’t audition for, but says he gladly would have, if asked. “I would have done cartwheels naked through crowded streets just for a shot.” Fortunately — or not, for his legion of fans — Tyldum didn’t make such demands.

Cumberbatch minces no words when he speaks of Turing, whose solving of Germany’s Enigma code saved countless lives and shortened the war. “Alan Turing is a war hero, a gay icon, a man whose work ripples on into our lives now,” he says. But Turing died in disgrace; he was convicted of homosexuality in 1952, and committed suicide two years later. “He forged a life so brief and tragic, yet so profoundly important, it shaped the world,” Cumberbatch continues. “And then he was thrown away by the very society he helped keep safe. Awarded with persecution and prosecution for being different.” It was only this year that Turing received a royal pardon.

“Imitation Game” reunites the actor with Keira Knightley, who plays his fiancee in the film. The two first worked together on 2007’s “Atonement,” though their characters were at odds in that movie. “Benedict is the real deal in every way,” Knightley says. “He’s constantly surprising you, and capable of anything. It wasn’t hard to be repulsed by him in one film and fall in love with him in another.”

Cumberbatch has played real people before, most notably Julian Assange in “The Fifth Estate,” a performance that didn’t sit well with the Wikileaks founder. Regardless, Cumberbatch says he always takes pains when portraying an actual individual. “I am determined not to carve out something two-dimensional,” he notes. “But you do have to make leaps as to what people are like behind closed doors.” In some ways, Assange was an easier performance, having so much footage to draw from. With Turing, there was virtually none, though Cumberbatch spoke to countless relatives and colleagues. “I had to use guesswork and forensics to put a lot of it together,” he admits. “It was a tightrope.”

Cumberbatch has not only received critical praise, but kudos from those he considers most important. “The biggest compliment I got was when his family came to the London premiere, and said (that watching the movie) felt like being with him again,” he says. “That’s the only review I need.”

Cumberbatch knows his next steps will prove crucial, being so prominent in the public eye, but acts at times as if he can will away his stardom. He chose to announce the news of his engagement to theater director Sophie Hunter in early November in a most old-fashioned way: having his parents place a formal announcement in the paid section of the U.K. newspaper the Times. “That’s how I would have done it if I wasn’t in the position I’m in,” he explains. “So that’s again me trying to normalize things.”

Still, he doesn’t want to sound ungrateful about all the attention, noting, “I don’t burn with regret at losing parts of my anonymity. I guess because this has happened to me in my mid-30s, not my 20s or teenage years.”

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Born and raised in London, Cumberbatch seems destined to have been a performer — his parents are actors Timothy Carlton and Wanda Ventham, who also play his parents in “Sherlock” — but he says they didn’t encourage him to become a thesp. “They wanted me to be anything but,” he reveals. “They afforded me a ridiculous education and all of their love; it was utterly selfless and self-sacrificing.”

While he toyed with being a lawyer, he admits he always wanted to be an actor. While attending the Harrow School at the age of 12, he made his stage debut in an all-boys version of Shakespeare’s “A Midsummer Night’s Dream” as Titania, Queen of the Fairies. It was when he played Salieri in a stage production of “Amadeus” at Manchester U. that he received his father’s blessing. Recalls Cumberbatch, “He came up to me after, in tears, and said, ‘I’m so proud of you. You’re better than I ever was or will be, and you’re going to have an amazing life doing this. I can’t wait to see it.’”

Cumberbatch surrounds himself with those who have known him all his life — before he became famous — to stay grounded, and he generally tries to stay off social media save for moments like his ALS Ice Bucket Challenge video, in which he gets doused repeatedly.

He has no desire to be known as a mere meme, and says the infamous photo of him with U2 at this year’s Oscars is not the kind of spontaneous moment he wants to pre-plan. “The amount of people I get asking me” — here, he puts on his best Valley Girl accent — “ ‘Omigod, can you photobomb me?’ No. I’m not a performing monkey. When it gets really banal and reductive, I’m like, let’s move on.”

Of course, it will be impossible for him to lay low as awards season ramps up; “The Imitation Game” is the great Oscar hope of the Weinstein Co. this year, and Cumberbatch is the crown jewel.

Though the actor is unproven as a leading man in a movie, Weinstein Co. co-chairman Harvey Weinstein was convinced of his talent when he first saw him in a production of “Frankenstein” at London’s National Theatre, where Cumberbatch alternated the role of the Creature and Dr. Frankenstein with Jonny Lee Miller. Says Weinstein, “His talent and skill are what every filmmaker dreams of working with. To watch how he not only captured the intelligence of Turing, but also the emotion behind the man at the forefront of computers truly did justice to the history and sacrifices Alan Turing made during a lifetime that was brief but immeasurably influential.”

For his part, Cumberbatch is promoting the film as much as he can, but says the most important thing for him is to concentrate on his work. “Thank God I have mechanisms and people and places to help me deal with this,” he notes.

His only qualm is that the time spent on publicity could distract from his work on “Richard III.” “If I go back and I’m tired on set or not enjoying the moment I’ve been given to play that character onscreen,” he says, “I’ll never forgive myself.”

As for what’s next, Cumberbatch has officially been announced as the lead in Marvel’s upcoming “Doctor Strange,” starring as the former neurosurgeon who protects Earth against mystical threats. Then there’s a new season of “Sherlock” set to arrive in 2015; along with other movies, including Warner Bros.’ “Black Mass” and “Jungle Book: Origins”; as well as “Richard III,” part of the BBC’s series of history-based Shakespeare plays under “The Hollow Crown” banner. What can audiences expect from his interpretation of the bloody king? “A few beheadings, some near-incestual marital arrangements, a hump and a limp,” he notes, before adding seriously, “Something very sexy, witty and dangerous.” The actor also will be heard onscreen this month, reprising his role as the voice of the dragon Smaug in the final “Hobbit” installment.

And, of course, there may be a trip to the Academy Awards in February. But this time, he vows, he won’t be asleep if his name is announced.