The musician Jess Mills gave birth on her bathroom floor, with a midwife shouting instructions through the phone and her mother, former Labour cabinet minister Tessa Jowell, holding her hand. In the odd way that the beginning of life sometimes comes out of nowhere, lazy and slow, before speeding wildly like a car in rain, after hours of labouring, Mills’s daughter was delivered in a rush by a young paramedic. Then, in the odd way that the end of life sometimes comes out of nowhere, 10 weeks after that Jowell had a seizure and was diagnosed with a rare brain cancer. She died the following May, holding her daughter’s hand.

I’ve learned a lot about grief, since Mum. You don’t get over a loss like this

It’s now 10 months later, and under the name SLO, Mills is releasing an album called Solace. Aged 38, she started recording it two years ago, before an enforced pause while her life combusted. On a rainy Tuesday, pregnant again, she welcomes me into the London flat she shares with her baby and husband, its rooms scented greenly with candles. And while we’re here to talk about the album, it’s quickly clear that the music (folky electronica, intimate and slightly ethereal) is a window, a way for her to talk about the many ways in which her world has changed. And motherhood, and cancer, and what grief is.

Family affair: Tessa Jowell greets a new baby in the family alongside her daughter Jess, right, and daughter-in-law Ella Mills. Photograph: DeliciouslyElla/Instagram

The lead single is titled Temporary Madness. “It’s about the miraculous process of healing, from a place of feeling irreparably broken to somehow becoming a functioning human being again,” she says, lightly. She’s sitting at her dining table with a large cup of coffee, and behind her is a framed photograph of herself laughing with Jowell, mid-chemo. “I’ve learned a lot about grief, since Mum. You don’t get over a loss like this. Grief is actually about learning to live with a broken heart.”

Boiling the kettle, Mills receives a text from singer Jessie Ware, who’s just had her second child. I ask what happens to a pop star when they become a mother and she chuckles darkly. “Well, I think the perception that once you become a mother you’re somehow less capable of operating at your optimum capacity is absolutely rubbish. Instead, you shift gear into a new dynamic space.” But do record labels see it that way? “Hmm. There’s a patriarchal hang-up of not being able to equate the female mother with someone who’s also powerful – it’s threatening to men who need that woman to be continually sexualised, and quite possibly not on their own terms.”

How does she feel then, releasing an album with another child to follow? “We’re in a very interesting time where all these patriarchal constructs that women have been forced to sit within for a very long time are crumbling. There’s definitely a collective-consciousness shift going on, which is powerfully reconfiguring the female in all aspects of society. The music industry is a place that is still framed by the patriarchal lens, more than the people within it would like to admit. But progress is about progressive optimism and believing in the change, so…” She shrugs.

Her album spans, she says, “A few different lifetimes. I find it strange, actually, listening to it because as well as charting that, it somehow represents continuity as well, and, though Mum’s not here, the things that stay the same.”

She pauses often when talking about her mum. Upon learning that her cancer was terminal, Jowell gave a speech in the House of Lords about the underfunding in neurological cancer. It receives only 2% of research budgets and has had no new vital drugs for the past 50 years. She received the first ever standing ovation in the upper chamber.

Early days: Tessa Jowell campaigns for Ilford North, February 1978. Photograph: M Fresco/Getty Images

So Mills will talk, then gather herself, then sip her coffee and continue gently, because while in theory she’s promoting her album, what she actually finds herself promoting, daily, is her mother’s legacy. Or, legacies – while the government has doubled its investment in brain cancer research in her honour, what Jowell will primarily be remembered for is her kindness.

She created Sure Start, she helped secure the 2012 Olympics, but in the tributes after her death it was her empathy and compassion that people talked about, regardless of their politics. That day Alastair Campbell called his colleagues to let them know she’d died, and admitted, “There was a Tory MP I spoke to who burst into tears.”

Mills is CEO of the newly founded charity ACT for Cancer, and the Tessa Jowell Brain Cancer Mission (TJBCM), a role she’s found herself taking owing to her experience as her mother’s carer and her campaigning spirit.

In 2015, Mills visited the Calais “Jungle” with her mum and came back changed. “I was haunted,” she says, “I can remember how I felt on the train that day. So I connected with these girls that had set up an incredibly dynamic, amazing organisation, Help Refugees. None of us had any experience in the sector at all. We were musicians, students – we just learned on our feet. And we created a new model for humanitarian response: now they work in 10 different countries and have helped more than a million people.”

She became a director of TJBCM and, in 2016, helped mount a legal challenge against the Home Office for failing to meet its commitment to provide sanctuary for unaccompanied child refugees.

When her mum got ill, Mills was able to transfer her campaigning skills. “The things that we saw as a result of her patient journey was a window into what thousands of other cancer patients go through every single day, so I’m now working on transforming patient access to the frontlines of therapeutic innovation.”

The day before we met, she’d cancelled all her plans in order to travel to Cambridge to help a friend whose son is being treated for cancer. To talk to doctors on her behalf and also just to be there. To sit. “You can’t underestimate from a patient perspective how tough it can be to rally doctors to do the things that you know are necessary. I was lucky in that I had the confidence and academic ability to understand the technical things that needed to be overcome with Mum, and we had amazing support to coach us as to what was needed. But, honestly, if we hadn’t had that, our mum’s process would have been completely different. Good advocacy for patients is life-saving work.”

‘Growing up, it was never just chit-chat around the dinner table’: Jess Mills wears dress and headband by Simone Rocha, matchesfashion.com

She smiles for the first time since sitting down. “Nobody can get the job done quicker, with more urgency and bigger ambition for the patient than the person who loves them most in the world.” TJBCM is developing a patient advocacy programme that will assist families that are navigating treatment, and it’s working: the Department of Health is currently implementing their commitment to share information and research across all stakeholders, from patients to government groups.

As Britain mourned Jowell, I wonder if it felt like strangers were imposing on her grief. “Mum gave so much to so many people, but the thing that she gave to us was absolutely unique,” she says. “Witnessing the outpouring of love didn’t feel that people were claiming her, because we could never have asked for more. We kind of knew that she saved the best for us.”

Yeah, growing up, some of my friends would be nervous about coming over for dinner

By us, she means her brother and their three step-siblings, a family that, it could be argued, are something of a cultural dynasty. Her father is lawyer David Mills, who in 2006 was at the centre of a scandal following accusations that he took a bribe from Silvio Berlusconi; her grandfather was an MI5 spy, and her sister-in-law is Deliciously Ella, the figurehead of the controversial “clean eating” movement. “Yeah, growing up, some of my friends would be nervous about coming over for dinner because it was never just chit-chat – never, like, ‘Lovely weather today...’” She says. Her eyes become very round. “Never.”

On Saturday, Mills woke up in tears. She’d been dreaming about her mum. “The trigger points are unpredictable and usually tiny things – a smell, or something my daughter does that my mum would go crazy for.” She pauses. She wants to explain what it feels like to have the worst thing she could have imagined come true, but then, to survive. “It’s like being out on the sea in a crazy wild storm, and there’s a slight transition where you go from being at the mercy of it, to the sun briefly shining, and you find a calm area to swim through. It’s like, people can only see colours that they have words for. If you don’t know what this feeling is you can’t possibly imagine it. It’s sadness and it’s heartbreak, but the infinite thing rooted in it is love.”

Jowell’s cancer was in the speech and language part of her brain, and towards the end she found it difficult to talk. “But there was one thing she’d always say very coherently, which I now see was really profound, and something I’ve learned to believe in completely. And it was so simple. It was ‘love forever’”.

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Fashion editor Jo Jones; hair by Tomomi Roppongi using Bumble & Bumble; makeup by Louisa Copperwaite using Mac