For an actress best known for her classical stage work, Fiona Shaw is suddenly all over our screens. In the last six months, she has popped up on TV in Killing Eve and Mrs Wilson, and in cinemas in Lizzie and Colette. Now she’s about to make a guest appearance in the feverishly awaited second series of Fleabag.

Born in Cork, Shaw left for London in her mid-20s and trained at Rada. Now 60, she made her name at the RSC and the National Theatre, where she shone in some of the great tragic roles, including Medea, Electra, Richard II, Mother Courage and Hedda Gabler. She has won two Olivier awards for best actress and an honorary CBE for services to drama, while her film credits include My Left Foot, The Butcher Boy and the Harry Potter series as Aunt Petunia Dursley. Now Shaw seems to have become Phoebe Waller-Bridge’s muse-cum-mascot, with roles in both her spy thriller Killing Eve as icy Carolyn Martens, head of MI6’s Russian desk who has “saved the world at least three times”, and now as the anti-heroine’s therapist in her comedy Fleabag.

With a broader, bouncier Irish lilt than you might expect – she’s usually on screen or stage with a cut-glass English accent – Shaw’s conversation roams around with inquisitive intelligence, while her eyes twinkle.

Even before our interview starts, she’s held forth between sips of tea on everything from modernist furniture (doesn’t like it) to London’s canals (loves them but doesn’t see the appeal of a houseboat); from the recent theft of her beloved bicycle (“I’m currently getting Ubers everywhere and it’s making me fat”) to the morning’s news of Cardinal George Pell’s conviction for child sexual assault (“The Catholic church has behaved diabolically. The pope says the devil is abroad but he’s not – he’s at home”).

How did your guest role in Fleabag come about?
I worked with Phoebe on Killing Eve and she was so helpful. I was frightened to do that role because it was a big departure. I was losing sleep worrying but thanks to Phoebe, I began to trust it. It’s almost anti-acting. You’re rarely saying what you mean. There’s lots going on inside Carolyn’s head but she says something else – that was new for me. But we finished it, it went down well and there was lots of celebration. Then I was directing Cendrillon [an operatic version of Cinderella] at Glyndebourne last autumn and Phoebe asked if I’d do something on Fleabag. I said no, because I was flat out doing the opera. But when it was over, she said: “I’ve just written another bit for you”. That time, I said [shouting] “OK, YES!” It’s a huge pleasure. I’m delighted to be in with the young crowd!

Were you a fan of the first series?
I thought it was marvellous. Fleabag is about Phoebe’s generation, certainly not about mine, and it’s wonderful. Phoebe reminds me of when I was 32 and doing very serious theatre. Plays that were utterly exhausting but I was also enjoying every minute of it. Now I see Phoebe enjoying every minute of her world. To watch her glorying in it makes me feel maternal. She’s having her moment: working so hard but relishing it and being radiant.

When we meet your character, you’re moisturising your forearms…
It’s an amazing introduction, isn’t it? An unseemly beginning, almost like a vet about to deliver a calf. Phoebe always does that, she never starts on a G-major chord. The first series opened on something quite distasteful too, if you recall [a notorious sex scene]. What was astonishing was that as we were filming, she’d rewrite along the way and it would be fantastic. To be acting and writing at the same time is so impressive. With Fleabag, Phoebe’s put her finger on something. It’s about a privileged young white woman but it touches on themes that are more universal.

You and Kristin Scott Thomas are both guest starring this series.
Phoebe’s quote [for the press release] was very funny. She said: “They literally begged me to be in it. One of them was crying.” I’m happy to cry and beg.

Do you wish you’d done more comedy?
I do. I also wish I’d played a therapist before. Put that in the Observer: I’m available for therapists! Because I think acting is about listening really. When I was younger, I played lots of active, physical stage roles. Suddenly on screen, I’m doing the opposite. The therapist in Fleabag never moves from her chair. Carolyn in Killing Eve is very still too.

When the second series of Killing Eve arrives in April, what’s Carolyn up to?
I can’t say much without giving secrets away. She’s steadying the ship initially and then … stuff happens. Is that vague enough? We left Carolyn with a cloud over her. I have no doubts about her but other people do. So we’ll see.

What’s been your favourite scene?
I loved it when Carolyn got drunk and flirtatious in Moscow. It’s great when someone so strict turns out to be very messy. It also shows the dexterity and freedom of Phoebe’s writing. Like when she suddenly decided that Sean [Delaney, who plays MI6 hacker Kenny] looked like me, so she made him Carolyn’s son. He wasn’t my son at the beginning.

Since Killing Eve became a worldwide hit, are you getting recognised more?
I’ve been in films that have taken off before. For years after Three Men and a Little Lady [in which Shaw played an amorous headmistress], I was mobbed by children. The ubiquity of Harry Potter meant I couldn’t even go to China without being noticed. Killing Eve has been similar. I can’t cross the street without people shouting at me from bicycles or cars. They all have their theories.

When people recognise you from Harry Potter, how do they react?
I had one experience recently when an air hostess was very impolite, then came up at the end of the flight and said: “I’m terribly sorry but I was frightened of you because my children watch Harry Potter all the time.” It’s funny. Children always gaze up at me in horror, while their parent doesn’t even notice and just drags them away. I’m sure they’d much prefer to see Dobby than me.

Are you going to be in all Phoebe’s projects, like her lucky charm?
Well, I just hope Phoebe comes a cropper if she doesn’t have me in something (laughs).

You also played an MI6 boss in Mrs Wilson recently…
MI5 in that, actually, but let’s not quibble about numbers! That was an interesting thing to do. Very brave by Ruth Wilson and an amazing gift that her grandmother had left this memoir [about Wilson’s grandmother discovering that her husband was a spy with multiple wives] but it also prised open a door to something wider. I can’t tell you how many people have come to me since with similar stories. I have a great friend whose parents met at Bletchley Park and her father had another family. I know someone else with a story of bigamy in north London. Postwar Britain was full of secrecy. It’s just unfortunate that Mrs Wilson came out just after Killing Eve, so it looked like I was typecast as spymasters.

You played Colette’s mother in last year’s biopic. How was that?
Lovely. Colette was such a part of my childhood because I read all the Claudine books. I was quite shocked to be playing her mother, though. I’d prefer to play Colette but that moment’s passed.

Are meaty film roles for mature women still rare?
Can you imagine how many grey wigs I’m offered in a week? Twenty years ago, I was in a film called The Last September with Maggie Smith and she said to me [slips into uncanny impression]: “The thing is, it only gets worse. There’s more harridans to play and you have to go on playing them.” I remember thinking: “Luckily, that’s miles away for me.” Yet here I am. But I’m not playing harridans, you see. There’s more on offer nowadays. There’s been a plainchant from women for the last 30 years that our roles are tedious to play. That’s why I spent so much time in theatre, because the roles are much better. The world’s changed and that’s a great joy.

Have the #MeToo and Time’s Up movements caused a cultural shift too?
The grip of the patriarchy is slipping and that’s to everyone’s benefit. It’s probably a relief for men as well. But I just think about the Rohingya women [in Myanmar] having their breasts chopped off at the same time as #MeToo is going on. Global oppression of women has got to be addressed. #MeToo is quite a local matter. I don’t mean it’s insignificant but it’s first-world problems in a world that’s much bigger. When #MeToo turns its guns on issues like the Rohingya women, I’ll feel it’s more relevant.

You’re an Irish woman living in England - is the motherland calling to you more as you get older?
It’s not. But my mother is still alive and going strong – she’s 92, still plays tennis and drives – so I haven’t got over my childhood yet. I’m still in that relationship where I go back to visit her. Nothing has changed in all these decades. I’m still in daughter mode, so haven’t thought beyond that.

Are you a workaholic?
Not now but I was for a long time. What changed? Age. Also I married and that calms you down. I didn’t really have a domestic life because I was always working. But I do have one now, which I love.

Don’t you also spend a lot of time in New York?
Yes, we also have a flat in Chelsea because my partner, Sonali Deraniyagala, lives there. She’s an economist and teaches at Columbia. So I’m enjoying all the theatres, cinemas and galleries there. I keep bumping into ghosts of myself from my youth. When I did The Waste Land in 1995, it was such an experience. The whole of Manhattan society seemed to come and see this tiny performance of a poem in a disused porno theatre on 42nd Street. Limos pulled up and ladies came in with chihuahuas. I was introduced to an incredible woman called Jean Stein, who later wrote a book about Hollywood called West of Eden. She was a doyenne of New York, took me under her wing and invited me to these dinners. You’d be sent the guest list in advance and get the fear because there’d be three Nobel prize-winners around the table.

Could you still recite The Waste Land?
Most of it’s still in there [taps head]. I was 35 then and I’ve thought about doing it again when I’m 65. I’m holding it tight until the right moment.

You used to live within earshot of London Zoo. Could you hear the animals?
Oh yes, especially on summer nights. I used to hear the elephants and it was beautiful. Sadly I moved from Primrose Hill to Islington and now I just hear buses growling.

What else does 2019 hold?
I’m about to do a film called Ammonite, directed by Francis Lee, who did God’s Own Country. It’s a historical drama about three real women – played by Kate Winslet, Saoirse Ronan and myself – during the fossil-hunting craze of the 1840s. We start shooting in a few weeks in Lyme Regis.

Fleabag series 2 episode 1 is released on BBC Three at 10am on Monday 4th March and airs on BBC One at 10:35pm that evening. The series will roll out weekly at the same times