Martin Freeman is not affable. This is according to Freeman himself, who has brushed off the label in the past. He’s not Tim from The Office; neither is he Dr Watson from Sherlock. He’s his own man, and when we meet in a hotel in Tribeca, New York City, there’s an unexpected spikiness to him. He’s not rude, far from it, but there’s an underlying edge and one that he’s all too aware of.

“I think I’m a pretty decent person,” he says. “I’m not horrible or unfriendly. But I’m my own person and I think sometimes people think ‘affable’ is going to mean ‘doormat’ or ‘just grateful all the time for any attention’ – and I’m not. I’m not grateful for the attention – quite the opposite, 80% of the time, dependent on the context. This is work and I want people to see my work, so it would be silly for me to be arsey in this situation. But when you’re out and about, people have an idea of whoever you are and I think they have a different idea of, say, Ray Winstone than they would of me. Maybe not now – word’s got around by now.”

Maybe it has, but being approached on the street isn’t something that’s going to stop any time soon. The 46-year-old actor has graduated from “guy from The Office” to “guy from pretty much everything” with roles in the small-screen take on Fargo and the big-budget Hobbit franchise, and a supporting part in the Marvel Cinematic Universe (MCU), cropping up as Everett Ross in Captain America: Civil War and Black Panther.

He’s here to discuss his lead role in Netflix’s zombie thriller Cargo, in which he plays an infected father trying to find a safe place in the Australian outback where he can leave his infant daughter before he “turns”. It’s a demanding role in a harsh landscape, with Freeman carrying a baby in the majority of scenes.

“We were lucky because we had two sets of twins and they were very well-behaved,” he says.

How does one get a baby to cry on demand without, say, poking them? “Pick the one who hasn’t had a nap yet, and for about 10 minutes she might be a bit arsey,” he says.

It’s another left turn for Freeman, who has had to work hard against typecasting. “If you’re not careful, you’ll never get out of it until you are too old to be a lovable Tim from The Office type – and then you won’t work again because that was your bit,” he says. “I wasn’t a comedian and I had a very deep hunger to not just do that.”

Freeman is now more likely to be recognised for his work in BBC’s Sherlock, a show that has gained global popularity and an intense fanbase to boot. As Freeman said in a recent interview for the Telegraph: “People’s expectations … some of it’s not fun any more. It’s not a thing to be enjoyed, it’s a thing of: ‘You better fucking do this, otherwise you’re a cunt.’ That’s not fun any more.”

The quote, along with the headline “Martin Freeman interview: Being in Sherlock isn’t always fun” soon caught fire, with his co-star Benedict Cumberbatch telling radiotimes.com: “It’s the responsibility of the storytellers to manage that, really. And I think, you know, it’s pretty weak to blame that on fans. You’re either along for the ride or not.”

Freeman claims he wasn’t aware of Cumberbatch’s comments, but remains annoyed with how he has been portrayed. “What the headline said, I didn’t say,” he says. “I didn’t say that Sherlock isn’t fun for me any more. I never said that. I was talking about different aspects of expectation of some fans. What I resented was it made it sound like I was saying doing Sherlock isn’t fun. There’s a lot of people I love on that show and that show has been extremely good to all of us, so yeah, I hate the pre ...”

He stops himself.

“I hate everything,” he says instead. “It’s rife with danger. The whole thing about it is when you open your mouth about anything, something is going to be taken the wrong way – and context is everything. You have to try and stay away from that stuff.”

Problems with the press have “gotten way worse” in recent years, he says. “In interviews you say things, but you wouldn’t print them on a T-shirt … We’re all living with that fear of being misunderstood.”

He’s just coming off one of his biggest press tours yet, for Black Panther, now the third biggest film of all time in the US. With a virtual mountain of tweets, memes and thinkpieces, the reaction has been staggering. “People have felt a need for it, and people don’t always feel a social need to go and see a film,” he says. “Certain members of certain communities feel that, ‘Yes, this is my film.’ I’m not making any claims that now America will change – of course it won’t. If America didn’t change under Obama, then ...”

He starts again. “You have to be realistic about what we do. We are making films, which are really important, but it doesn’t change everything. It’s just very nice to be part of what has been a very important film to a lot of people.”

As for his future in the MCU, he says his initial call from Marvel chief Kevin Feige was for “Civil War and a couple of Black Panthers”, so it is likely we will see him in Wakanda again.

And while we’re on the subject of satisfying fan fervour, how about more Sherlock? “I think there could possibly be more with it – but at the same time, the way that the last one ended did feel like, if there’s more, it won’t be for a wee while,” he says.

Despite a range of roles that have taken him to the US, Freeman still lives in the UK. As a passionate Labour supporter (he even fronted a party political broadcast in 2015), how has he found life post-Brexit?

“Here’s the thing about Brexit,” he says. “Yes, I am generally remain, but the problem is that it’s so polarised that it’s too easy to say … Well, a big problem with the remain campaign was that it asked, ‘Are you for remain or are you a racist?’ There are traditional leftwing Labour reasons to be Brexit. Tony Benn – patron saint of the left in our country. He was anti-EU. Whatever you think of a lot of people on the left in Britain, they’re not little Englanders and a fair amount of them were not for the EU. So that to me was a silly tactic for the remain campaign.”

But enough about customs unions, non-tariff barriers, taking back control and so on. What next for Martin Freeman? “I’m reading a lot of scripts,” he says. And if some of them only add to his reputation for affability … well, just don’t tell him that to his face.

Cargo is on Netflix from 18 May.