Shadow home secretary Yvette Cooper has long been committed to women's rights. But where does she stand on this week's row over surveillance legislation? And can we get her to admit she'd like to lead Labour?

As we sit down in Yvette Cooper's corner office, which looks directly on to the Houses of Parliament, an aide brings two mugs of tea, one with a line of red and another with blue. The shadow home secretary passes me the red one, laughingly denying that she is making a party political point. "That one's got more in it," she says, before sticking to the party line for much of the next hour.

It has been a big week for Cooper. She agreed to this interview to discuss Labour's plans to draft landmark legislation on women's safety, but that was before the inquiry into child abuse was announced, and before deputy Labour leader Harriet Harman lobbed a bomb into the party hierarchy, insisting that the Gordon Brown era was marked by sexism and inequality. We finally meet just before the highly controversial emergency surveillance legislation is announced following secret meetings between the three parties.

I've long been a fan of Cooper. Her focus on the impact of the austerity agenda on lower-income women drew attention to the coalition's tin ear on the issue, and her continuing commitment to women's rights has been impressive. She is admired in her party for being smart, sensible and no-nonsense, and any suggestion that she isn't "clubbable" can be attributed to the underlying sexism of a place where ministers who prefer putting the kids to bed rather than drinking with colleagues are deemed a bit odd.

And yet, even those who regret that she didn't enter the race for the Labour leadership in 2010 find it hard to know what she really stands for. Like many who came to power under the Blair-Brown aegis, she has learned to say nothing that hasn't already gone through the "will this win votes" wringer. It can make for a frustrating interview, particularly when the sensitive issue of surveillance comes up. Admittedly the timing of our interview is awkward, but trying to get Cooper to spell out where she stands on the key issue of personal privacy v state security defeats me.

The day after we meet, Labour controversially supported emergency legislation that shores up government powers to require phone and internet companies to retain and hand over data to the security services. At Westminster on Thursday, Cooper raised concerns about the rushed legislative process, but veteran Labour backbencher Tom Watson was critical of her role: "For the people who care about issues of civil liberty and privacy, her failure to secure a proper public debate about state surveillance in the internet age is very disappointing. In the end, character is judged on behaviour, and on this she's been found wanting."

Cooper defends herself, saying Labour successfully negotiated a sunset clause, ensuring that the bill will expire in 2016, forcing a proper debate at that point on the balance between surveillance and privacy.

Though she may not convince everyone on surveillance, her focus on the needs and concerns of women is genuine and much-admired. Straight after graduating in 1990, Cooper won an award to travel to the US, where she researched abortion rights. Above her office fireplace is a huge portrait of suffragette Emmeline Pankhurst made up of hundreds of tiny portraits of inspirational women – Cooper enthusiastically points out Aung Sun Suu Ki and Mo Mowlam. On the wall is a war poster urging mothers to vote Labour.

For those who would like to see the Labour party led by a woman – some 40 years after the Conservatives – Cooper has long been the most likely contender. In many ways, she ticks all the boxes. The daughter of a trade unionist and teacher, and granddaughter of a miner, she went to a comprehensive school in the south before being elected in the tough northern constituency of Pontefract and Castleford. In between she got a first at Oxford and did work experience with former Labour leader John Smith and US president Bill Clinton before a brief stint as a journalist. But she ruled herself out of the last leadership election in 2010, citing the ages of her three children, while infamously allowing her husband, shadow chancellor Ed Balls, to stand instead. She concluded an impassioned defence of her decision by saying, "As for future leadership contests, who knows … " prompting every single interviewer since to ask if she might change her mind. And I'm no exception.

First I ask whether it matters that the Labour party has never been led by a woman. "Of course you want to be able to have women leaders in the Labour party, just like you want to be able to have women in every position, of course that's a good thing," she begins. "But equally you can also have a Labour party standing up for women. And Ed Miliband has made a conscious choice to have a balanced shadow cabinet … He also raised the issue of not having enough women in cabinet at PMQs."

OK, I persist – if he weren't doing the job, would you want to do it? "I want to be home secretary," she says so quickly that I laugh and ask why she can't just admit to being a little bit ambitious for the top job. "I want to be home secretary, that's the job I want to do," she repeats. "I think Ed is doing a great job and we've got to do everything possible to make sure that Labour wins the next election because, frankly, if we don't, the consequences of five more years of the Tories [will be] chilling."

Others, such as former cabinet minister David Blunkett, have suggested that a Conservative win at the next election could keep Labour out of power for a generation, ending the chances of all of those who grew up in the Blair-Brown years ever returning to power. Cooper screws up her face: "I'm just not interested in any of that stuff. We are now nine months from a general election. There is a huge amount at stake about the future of the country. It's not about parliament and it's not about individuals – it's about the future of the country. That is the most important thing, and it's what we're all campaigning for."

I'm not entirely convinced. Timing is all in politics, but it's hard to imagine that, as her children get older, she won't be tempted to stand if the opportunity arises.

Yvette Cooper with her husband, shadow chancellor Ed Balls, in 2008. Photograph: Brian Mackness

Cooper is much more comfortable having a go at the Conservatives over their record on women. "The problem in the Tory party," she says, "is that David Cameron not only has a problem and a blind spot with women, but that there is no one in the system who challenges him on it." What about Theresa May, her opposite number in the Home Office? "I think she just does her own job and her own thing, really." Besides, she says, "I don't get the sense that No 10 listens to her hugely."

It's a bit odd, not to mention sexist, that anyone would expect the two female home secretaries to have much in common, and yet many do, as though they must compare shoes as they walk into the chamber. "I can't say I know her very well," says Cooper. "Theresa doesn't talk much to, or rather doesn't do much associating with, Labour MPs or … " She tails off, before adding that "to be a senior woman in the Conservative party is hard".

There's a policy focus to her criticism. May might have been dealing with the child abuse inquiry and battles with her own colleagues about extremism, but she has "not done very much about violence against women, and didn't do so much when she was women's minister," she says.

In contrast, Cooper wants to put violence against women and girls alongside the more traditional core Home Office priorities of terrorism, policing and immigration.

Her proposed bill would appoint a new commissioner with a remit to inspect and report on women's safety – including the traditionally difficult-to-prosecute issues of female genital mutilation and forced marriage. "We must not turn our backs on victims of violence and fail to respond when the criminal justice system lets them down."

One of Cooper's biggest political assets, when compared with her identikit male counterparts, is an ability to appear like an ordinary human being from time to time – most often when talking about juggling work and home commitments. She is funny and engaging on the demands of managing three children while married to a fellow politician. On the evening after we meet, Cooper had double-booked herself with a gala dinner and a school concert and was looking forward to a "mad drive" between the two. "I'm not going to miss the concert because it's really important, so I might miss the starter instead. I've done things before where I've just had to not eat food to fit it in."

Crediting her mother for helping to fill in the gaps where the couple's madly competing cross-country schedules collide, she says: "You have to just sort of assume that things will go wrong, 'OK, I'm late again and I haven't done everything perfectly.' There is always chaos. You just have to be comfortable with a certain level of chaos."

Cooper was the first minister ever to go on maternity leave, and has written about how badly her own civil servants treated her when she did so. Asked about her experiences of sexism in Westminster, she recalls just recently telling a Tory MP after a vote on a humid day that she felt "all hot and sticky". "He just started giggling and said, 'Oo-er, I can't say anything to that,'" she groans. "It was really weird."

Among the more shocking details of Harriet Harman's speech this week, in which she complained of being passed over for the role of deputy prime minister, was the fact her own colleagues had mistaken her post-baby bulge for the baby, and reported her for carrying it through the lobby, while party whips leaked details of her mastitis to the press.

Has Cooper, an MP for 17 years since the age of 28, been subjected to the same treatment by her own party? There is a long silence before she says: "I have a tendency to just keep on going and not really notice."

What she has noticed, she says, is how many of Labour's equality programmes – all-women shortlists among them – have helped lead to a higher percentage of female Labour MPs than other parties. The shadow cabinet is 44% women, compared with 22% in the government. "When I started, there was still a rifle range in parliament. Now there is a nursery on the floor directly below."

She credits Harman for championing many of these changes, and is a huge fan of her older colleague. "Harriet has always been ready to speak up, even if she ruffles feathers," she smiles. "She has been in politics a long time, and what she has done is show that it's possible to smash through glass ceilings all over the place, and show other women that it's possible."

As we leave her office, a half-naked child wanders into the corridor, and then the lift stops outside the in-house nursery for Jenny Willott, a Liberal Democrat whip, who is pushing two small children in a double buggy. It's tempting to see this as a sign of real change, and yet it fails to erase my sense that, so close to a general election, it is very much business as usual in Westminster. True honesty is still in short supply in a system riven with party politics and a straitjacket electoral cycle.

I'm reminded of something Cooper said earlier, when talking about the pressures of this time of year for working parents, with its surfeit of plays and, "Oh God, not another school fair". Cooper describes making her excuses at a meeting of women's organisations by saying she had "another commitment" – really, she was going to sports day. "Of all of the meetings to which I could have said, 'I'm going to school sports day', that was the one, and still I didn't."

• This article was amended on 16 July 2014. An earlier version referred to a straightjacket, rather than straitjacket, electoral cycle.