Cafe Luxembourg, the closet-sized New York bistro Bernadette Peters chooses for our lunch, opened in 1983 and seems almost unchanged. That also goes for Peters, although she has been in business longer. As a child actor, she made her New York debut more than 60 years ago in a revival of The Most Happy Fella and has rarely been off the stage or screen since. A gift to musical comedy, in some very dishy wrapping, she synthesises sex, sophistication and emotional nuance. Has another Broadway baby made it to the cover of Playboy? “Just Ethel Merman,” she quips.

At 71, she looks enviably young, in a way that suggests an unholy pact and/or a virtuoso dermatologist. (Her explanation? Good genes.) On the Cafe Luxembourg computer, the reservationist has placed a star next to her name and, really, who wouldn’t?

In January, she took a trip to Tanzania and Rwanda – she shows me a smartphone video of a munching gorilla. “It was so hard to come back,” she says. “To come back, it was like, ‘Oh, I sing and dance now?’” Well, yes. Peters has been giving concerts – some Rodgers and Hammerstein, some Jerry Herman, a lot of Stephen Sondheim. She opens each show with Let Me Entertain You. “We’re here together,” she says, explaining why she chose that song. “How about this? Let’s have fun.”

Peters is dressed in a slinky black shirt and overlapping necklaces, her Titian ringlets bouncing. She shows pictures of her dogs – Charlie, her “shaggy dog”, and Rosalie, her pit bull. She recommends lipsticks such as Nars’s Lana. But she doesn’t like to explain what she does best, which is to imbue a song with so much imagination, experience and unadulterated feeling that it sounds as if it was written for her. (In fairness, sometimes the song was.)

“You keep it all secret,” she says. “You keep everything a mystery.”

Before meeting her, I polled some friends: what should I ask? Half of them wanted to know about her work with Sondheim. The other half wanted to know about those ringlets. One outlier wanted to know what it was like to date Steve Martin, her co-star in The Jerk and Pennies from Heaven. (“He can be wild and crazy, but basically, he’s a very serious guy.”) Where to begin? With Sondheim, she decides.

James Lapine had invited her to workshop Sunday in the Park With George. At the first rehearsal, they met in “this awful room where the acoustics are really bad” and Sondheim sang through the first act. “Oh my God, that was so beautiful,” she says. He had her open with the title song, which requires extraordinary breath control. “My voice had to be open and ready to go and – bam!” she recalls. “It was the beginning of a lovely relationship.” That relationship has taken her through Into the Woods and into revivals of Gypsy, A Little Night Music and Follies.

“A lot of his songs that I love are the ones that are inspiring. They help you in life,” she says, quoting a song from Sunday: “Stop worrying where you’re going / Move on.” That song taught her to embrace change. Doing Gypsy, she said, a show she had toured as a child actor, “was like doing therapy”.

What is it like to work with Sondheim? “Well, you’re in awe, you know, so it’s hard to focus.” Sometimes, during previews, she would make a suggestion. He would think about it, she says, really think about it. “And he would go, ‘Mmmm, no.’”

“I always trusted him, but he trusted me to put the way I saw the character and my passion into it. I had to interpret it.”

When Peters first reads a script, the character can feel far away, dissimilar. But she tries to connect. She quotes Sunday again, “You know, ‘Connect, George. Connect!’” and the character moves closer. “Every role I’ve played, I’ve thought, ‘That’s me. That’s me!’”

With that in mind, I suggest a game of free association and she consents, though as it turns out, she is not especially free with it. I name a character and she responds with the first words that come to her mind. And here are those words (you’ll have to imagine the long pauses):

Emma in Song and Dance: “I love her! It’s a girl trying to get through life, trying to find love in New York.”

Dot in Sunday in the Park With George: “Dot is trying so hard and trying so hard and trying so hard get his attention. And it was heartbreaking, really. But, it’s a love story.”

Then she more or less gave up. “I never look at them from the outside,” she says. “I’m always like: start the show, start my journey … I get so invested in stuff. I get invested in the shows, I get invested in what the character goes through every night.”

When Peters is working, she organises her life around the show, arriving at the theatre hours early, trying to avoid anything that might distract her. “You have to be very regimented,” she says. “You have to make sure you get eight hours of sleep. You can’t go to loud restaurants – you’ll lose your voice. You can’t talk on the phone too much. But the payoff is so, so beautiful. You make your instrument ready to go, ready to receive.”

In case you were curious about her physical beauty, let’s start with the hair. She straightened it until her late 20s, she says, then she let it curl. These days she swears by a leave-in conditioner from Mixed Chicks, with occasional help from a curling iron. Her skin she owes to her Sicilian genes, a devotion to the Mediterranean diet and the fact that she never learned to swim, “so I never went in the sun”. (A few years ago, she did learn to put her face in the water, for a cut scene from the movie Coming Up Roses.) She used to use a moisturiser from La Prairie, but the company changed the formula.

I ask if her prettiness ever seemed limiting, those decades of reviews that compared her to Betty Boop and kewpie dolls. The question surprises her. “I’m loving this!” she says. “I never thought I was pretty. That’s so funny.” But she acknowledges that people didn’t always take her seriously. “I always thought, ‘That’s OK, because one day they’ll see that I can do that.’ I used to have the best attitude.”

Her attitude still seems pretty good and, while some actresses struggle to find rewarding work after 40 or 50 or 60, this hasn’t been her problem. Peters recently stepped into the revival of Hello, Dolly! just as she was winding up a starring turn on Mozart in the Jungle and an arc on The Good Fight. She doesn’t have any dream roles, but she likes surprises. “Things are brought to me,” she says. “I think, ‘Oh, this would be nice to do. This might be exciting.’” (A new enthusiasm: Beyoncé. She’s been listening to her at the gym. Will Single Ladies work its way into her set list, between Children Will Listen and Being Alive? Not any time soon.)

She hopes to catch Follies when she is performing her concert in London, once she is over the jet lag. Is it ever strange, to see someone else play a part she has inhabited, to hear someone else sing the songs that are still in her repertory?

“It’s a relief,” she says. A few years ago she saw the revival of Sunday in the Park With George, and watched Annaleigh Ashford play the role Peters had created. For years she had wondered what it would be like to let someone else tell that story, to let someone else’s passion move an audience to tears. Now she doesn’t have to wonder. “Oh my God,” she recalls. “Now I know what this is all about.”

• Bernadette Peters is at the Lyceum, London, on 10 June. Then touring the UK until 18 June.