There is a poacher turned gamekeeper feel to interviewing Jessie and Lennie Ware over lunch. It’s usually them, in their double-act Table Manners podcast, doing the grilling – while doing the grilling. The 34-year-old singer-songwriter and her irrepressible mother have had all sorts of guests for supper: Sadiq Khan, Yotam Ottolenghi, Ed Sheeran (with whom Jessie wrote her biggest hit, Say You Love Me). They fed Cheryl Tweedy so well she had to leave with her jeans undone. They made Nigella’s pastry for Nigella. When George Ezra came over they had to order takeaway after they produced inedible short ribs. For this interview, though, they are away from the kitchen table. We are at Leroy, in London’s Shoreditch, a Michelin-starred sharing-plate bistro.

When we meet, Jessie is eight months pregnant (she has since had a son, her second child, a home birth celebrated on Instagram). I ask her about cravings as we study the menu. She confesses only a hankering for all the cheese she is not allowed. Otherwise, she has always been “a greedy cow” she says, and happily she now has more excuse than ever. She orders accordingly: smoked mackerel, burrata, cockles, jerusalem artichoke, monkfish. A brief pause. “And can we get the celeriac remoulade, the anchovies and ricotta. And we better have the pheasant. Is that going to be enough?”

Lennie has arrived a bit late and breathless after a walk from the tube – “has my makeup run?” – and mother and daughter are immediately into the familiar routine that makes their podcast sound like eavesdropping at a proper family table: constant low-level bickering, fencing for weakness, alive to old annoyances, all slathered with a generous helping of love. After Lennie separated from her husband, the longtime Panorama reporter John Ware, when Jessie was 10, Lennie brought up her three kids mostly alone. You sense they are a strong team. And their chosen sport is food.

Jessie mentions she has been seeing an ayurvedic practitioner, which causes her mum to roll her eyes, and has been advised to avoid gluten, though she is not gluten-intolerant.

Another eye roll: “Jessica, you are not doing that.”

“Mum, I am.”

I’m the father of two voluble teenage daughters so I slide easily into my habitual Mr Bennet role as they chatter, prompting occasionally, mostly letting them get on with it.

The podcast developed naturally. Lennie had always been in the habit of doing big Jewish Friday suppers for her children’s friends – “I’m not very Orthodox, but it was fun – I’d make chicken soup and matzo balls, which everyone adores.” When those friends became a bit more famous after Jessie brought out her first album, broadcasting those family events seemed like a jolly idea.

“I thought I would just be helping her with the cooking,” Lennie says, smiling, with mock humility. “I am a selfless Jewish mother.”

“I think she was hoping it might lead to me getting the This Morning gig alongside Phillip Schofield,” Jessie says.

“But of course, the wonderful thing is that people adore her.”

Table Manners has had 3.5 million listeners, and every episode gets an immediate 60,000 downloads. Lennie, who has worked for 40 years as a counsellor and social worker in family law cases, has found a degree of unexpected fame. “It’s what Jessie calls my whining voice that gets recognised,” she says. “I was trying to get on the tube the other day, and I was saying to this woman: ‘Bloody hell, we are not going to get in, are we?’ She said: ‘You aren’t the lady from the podcast are you?’”

“Exactly, she recognised you because you were whining,” Jessie says.

“And then I was in the sea in Greece, and somebody else recognised my voice. I have a big fan base.”

“People get annoyed when we talk over each other,” Jessie says. “And when you talk too much about your son, the doctor.”

Alex, that son, has a walk-on part in the podcast, mainly wielding the ice-cream maker that he received as a barmitzvah gift. “Alex is studying anaesthetics at the Royal College,” Lennie tells me.

Mother and daughter eat out as much as they can, partly to nick ideas, Jessie says. She examines the plate of ricotta and anchovy to see how they do it. “Is that a coriander seed or buckwheat?”

“Don’t panic, but it’s probably a tiny bit of gluten darling …”

“Oh, piss off.”

They are fretting a bit about what to cook for the next podcast, two days later. They have Hayley Squires, the star of I, Daniel Blake coming.

“Do you want to do that cardamom creme brulee we talked about?” Jessie says.

“No, I get nervous doing a new thing.”

“The only new bit is crushing some cardamoms into it.”

“If it’s so easy why don’t you spend your evening crushing cardamoms?”

“Well, two reasons. One, I’m heavily pregnant. And two, I’ve got Gilmore Girls to watch.”

I ask Lennie whether she knew Jessie would be a pop singer (her other daughter, Hannah, is an actor). She explains how Jessie, who had started out as a journalist on the Jewish Chronicle, was about to start a law conversion course when she was offered a recording contract. “She was always singing, always in the school shows,” Lennie says, “so I told her: ‘Why not give it a go?’”

It was a hard choice, Jessie says. “Part of me wanted to follow in my mum’s footsteps and go into family law.”

“I have loved being a social worker, but it is a great regret to me that I didn’t do law,” Lennie says. “I was the first in my family to go to university, and I went into social work really to avoid going into M&S management, which my mum had her eye on for me. I think she liked the idea that you got half-price food at the end of the day.”

The jerusalem artichoke arrives and is so good that they wonder if they should do that for the supper on Friday.

“In some ways, the podcast has really helped my music,” Jessie says. “It gives me less time to stress about all that mad world, which can consume you.”

She is determined this time to take more time off with her baby. She shares the childcare with her husband, Sam Burrows – the couple have been together since they were both 18, but the first time around she was too determined, she says, that the birth of her daughter would not interfere with her career.

Jessie took her to America when she was three months old. “I was making my album Glasshouse,” she says. “Mum came to help. She was at Pete Tong’s place, looking after the baby while I worked.”

“I got into a mess with all the formula milk powder and then I got worried people would think it was cocaine,” Lennie recalls.

A bowl of celeriac and chestnut soup arrives, which Lennie declares delicious, and some cockles, which she doesn’t fancy.

“I’ll have a couple,” Jessie says. “It might bring on this labour with any luck.”

“Oh, Jessie, don’t say that.”

I mention something I read recently that suggested mothers have better conversations with their daughters once those daughters become mothers themselves. Is that their experience?

“I think I have more respect for my mum now I have a baby,” Jessie says.

“I think we get on worse. I have to be so careful what I say.”

“What are you talking about?”

“She thinks I am too liberal a grandmother,” Lennie says.

“She thinks I am a sergeant major.”

“I think if my granddaughter wants jelly why can’t she have jelly?”

“Because she hasn’t eaten her dinner.”

“Jelly never did you any harm.”

Jessie says her next album will be slightly less personal than Glasshouse, with more “a seductive late‑night feel”.

“We don’t want to turn into the Kardashians,” she says.

The question of puddings is raised.

“No, no, nothing for me,” Jessie says.

“Oh, you have to have a pudding Jessie.”

“How could you, Mum, after all that pheasant?”

Lennie orders the rhubarb with herb ice-cream.

“We’ll share won’t we?” she says to me, conspiratorially. “And we won’t let Jessie have even a taste …”

Series 6 of Table Manners is available now