Tracey Ullman might just be the most important woman in the history of British comedy. She was the first to be offered her own sketch show – both here and in America – the first pop star to put a politician in her music video, the first foreign female comic with a hit show on US TV and the woman who discovered The Simpsons. You’d expect her to be name-checked by every new sketch troupe or home-grown comedy hopeful. Instead, she admits with a wry grin, no one even recognises her on the Tube.

“I haven’t worked here in a while,” shrugs the 56-year-old comedian, sounding just like the same south-London girl who first entranced the nation in the 1980s. “So I am always anonymous, always sitting on the Tube next to people. I go to the BBC and people think I’m Julie Walters. I sign in and security goes, ‘Oh, I know who you are, Miss Walters!’” Ullman laughs then slips into a perfect Julie Walters impersonation. “And I go, ‘Oh, crikey, thanks very much!’”

That may be about to change. Having spent the past three decades living in Hollywood, Ullman is back on primetime British TV with Tracey Ullman’s Show, a rollicking blend of impressions – from Angela Merkel to Dame Maggie Smith – and wonderfully silly character comedy, including an entirely topless female MP (the breasts are prosthetic, she points out) and a character simply known as Karen, a drug mule.

Maggie Smith and Judi Dench are there for a reason: “I think we have wonderful older actresses in this country who aren’t afraid to look old – in fact, when America needs a mother they have to import one of ours because American women have done so much to their faces, they don’t want to look over 60,” she quips.

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But what of Karen the drug mule (above)? The on-screen conceit is that she has returned to the UK after 30 years in a jail in the Far East. Is Karen’s awe at the changes she finds in today’s Britain based on Ullman’s own experience?

In fact, Ullman explains, she’s always travelled between the US and the UK to visit her daughter Mabel McKeown, a 29-year old charity worker and Labour Party activist who lives in London, and, until recently, Ullman’s mum Doreen. The first scene in the show – with young Tracey dancing in her mother’s bedroom – is a tribute to the woman who brought her up alone from the age of six, and who taught her to be funny. Doreen died last year, in a fire at her Buckinghamshire home.

“I didn’t have the ideal start to life with my dad dying,” says Ullman, referring to the early death of her father when she was six. He had a heart attack while reading her a bedtime story. “So it was a bit tough for my mum. I used to put on shows to cheer her up. That little scene in the bedroom dancing around – that was where everything started for me.”

When her mum remarried an erratic and occasionally violent man, Tracey retreated further into performing, learning mimicry to avoid playground bullies as her family criss-crossed the country and she went from new school to new school. All of which stood her in good stead when, as an ex-dancer working in a paper products company, she invented a born-again Christian chanteuse called Beverly for a new character night at London’s Royal Court Theatre. BBC comedy chief Paul Jackson and Lenny Henry caught her turn and persuaded her to join the cast of 1981’s A Kick Up the Eighties – which introduced Rik Mayall and Robbie Coltrane and helped pave the way for the alternative comedy of Dawn French and Jennifer Saunders, the Comic Strip, The Young Ones and Blackadder.

“When they asked me to do it, I really didn’t know,” she recalls. “I remember saying to them, ‘What do the girls do, wear bikinis and get their bums pinched by Benny Hill?’ But Paul said it doesn’t have to be like that. He knew that to get people like myself, or Dawn and Jennifer, he had to let us do what we wanted to do. I wasn’t a traditional-looking girl, let’s face it! I was a bit odd...” she laughs.

From there, her career exploded – the BBC offered her a show, and she chose to work with Lenny Henry and David Copperfield in the ground-breaking Three of a Kind, which avoided every sexist and racist stereotype in a way that still seems cutting edge today. She also signed to Stiff Records and released a string of hit singles, including a cover version of My Guy by Madness (see below). The song’s video starred then Labour leader Neil Kinnock, and opened the door for politicians to turn up on Have I Got News for You and One Direction in Comic Relief spots.

“Neil Kinnock is a good friend,” she explains. “And yes, I think generally I have been a Labour supporter and member for years and years.” Would she ask Jeremy Corbyn to join her on screen? She’s a little taken aback. “I don’t think a politician would do that now... and I’m not overtly political.”

Although one sketch in the new series takes place in a library facing closure thanks to funding cuts, which sounds like a political point, albeit of the most genteel kind. “I do live near a library that’s going to become a private house, and it just breaks my heart, but really I just wanted to tap dance in an unlikely place.”

At the height of her pop/sketch stardom, she suddenly disappeared – heading to LA with her new husband, Birds of a Feather producer Allan McKeown, who persuaded her to try out her talents there, even though she didn’t have any promises of work. The gamble paid off. Together they produced The Tracey Ullman Show for Fox, which discovered Dan Castellaneta – who went on to be the voice of Homer Simpson – and included cartoon skits that evolved into The Simpsons themselves. In the 90s, Ullman’s show moved to HBO for some one-off specials and Tracey Takes On... an Emmy-winning topical show. In 2008, she shifted to Showtime for the deeply satirical State of the Union, taking on the politics of the Bush era.

Because her US shows didn’t get an airing in the UK, even the smartest comedy fan under 50 has barely heard of her – but in the US, the shows made her a legend. “Tracey is absolutely beloved here,” explains Bruce Wagner, who wrote for Ullman’s State of the Union. “This was borne out daily whenever we walked in the streets. Black people were always particularly exuberant in their voiced praise. Tracey is a chameleon and her impersonations of all races, ages and walks of life are purely egalitarian. One of the eeriest things that ever happened was that we once went to dinner with executives from Showtime and Tracey started doing an imitation of me. It was so accurate that the near impossible happened: I saw myself as others do. It was unsettling, to say the least.”

Meryl Streep with "soulmate" Tracey Ullman

Perhaps the most famous member of Ullman’s US fan base is one Meryl Streep. “Tracey Ullman shot like a sparkler into my vision on the set of [1985 film] Plenty,” says Streep of her former co-star, now her friend.

“I thought I’d found my soulmate: a restless, silly, musical wild mimic, with a dark underlay of sangfroid. I loved her on sight. I was absolutely blackout-shocked to discover she was barely 23 years old (I was 33) and I still feel like her (slightly cowed and aspirant) little sister. She hasn’t changed in spirit or matter in all these years that we have remained close. I have no objectivity, clearly, but I am not alone in this assessment of her!”

After 30 years of marriage, Ullman’s husband Allan died in 2013, aged 67, from prostate cancer. His death spurred her return to the screen – and the UK. “He’d be thrilled that I’ve finally done it all and got a team together,” she says. And no doubt he would be thrilled to discover, as she was, that the old bow-tie-wearing BBC execs of the 80s had been replaced by young, smart women, like BBC controller of comedy production Myfanwy Moore.

“Before I met her I thought, because of her stardom in the States, Tracey might be a bit of a spoilt LA woman – wealthy, pampered, may be even a bit of a dragon,” Moore admits. “But she was the total opposite – very British still in sensibility and humour, casual, funny, thoughtful. She was a female icon for a whole generation – a great comic actress and impersonator, with perm and ra-ra skirts like a fun and funky role model, at the heart of popular culture. She is one of those rare things – a multi-talented performer who has a mainstream sensibility with a truly satirical streak. I’m glad Blighty has got her back.”

Ullman, in turn, is glad to be back. “It’s like coming home doing my voices here,” she explains, “It’s in my roots, in my soul. It’s like being able to play football or play the piano, you suddenly just key into someone’s accent – being born in England, as soon as someone opens their mouth, you know where they’re from, what they aspire to, where they went to school. Our class system was embedded in me from an early age, and it still prevails.”

It’s fair to say she doesn’t generally get the respect from the younger comics – fan boys mourning Rik Mayall’s death rarely mention their I collaboration, and she’s not part of Dawn French’s recent autobiographical show even though, in Jennifer Saunders’s words, “Tracey was a lesson in how to act funny.”

One star who does acknowledge their debt to Ullman is Catherine Tate. “I think she’s wonderful,” Ullman enthuses. “She sent me a lovely good luck letter when I started filming. I really appreciated her support, because she’s brilliant... and I love sketch comedy. There haven’t been any for a while so it seemed like a good chance to have a go, and it’s what I love to do. I guess I’m still putting on a show in my mum’s bedroom with her nightdresses on and I hope I’ll keep doing it until I’m 90...”

Tracey Ullman's Show begins on BBC1 today (Monday 11th January) at 10.45pm