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Jo Swinson is in high spirits. After years in the wilderness, her party, the Liberal Democrats, are popular again, topping a national poll this month for the first time since 2010. At Tuesday’s anti-Trump protest in Trafalgar Square, Swinson was surrounded by new fans, telling her they had just joined the party.

“In some cases it was because they had seen me on Question Time,” she says proudly, neatly folding up her on-brand Lib Dem yellow leather jacket. Swinson, currently the party’s deputy leader, is running to take over from Vince Cable when he steps down in July. She’s just returned to her Portcullis House office after a morning campaigning for the Peterborough by-election and explains that she put on shoes she can pound the streets in this morning: silver brogues that are “sensible but still interesting”. Her work paid off — the Lib Dems increased their hold there to 12.3 per cent, taking votes from the Tories and Labour. She also appears to have memorised the flow chart that’s pinned above her desk next to a poster of Hillary Clinton. Boxes read “How we got here” and “We are the solution”.

Swinson’s party surged into second place behind the Brexit Party in the European elections, beating Labour in Jeremy Corbyn’s Islington constituency. “People are flocking to us because they know we are going to stand up against a Tory Brexit,” she explains, still in campaign mode, a Vote Lib Dem sticker unfurling from her black T-shirt. “We’ll fight for our place in the EU, because that’s how we improve prosperity.”

The EU referendum gave Swinson a new sense of purpose. When she was elected MP for East Dunbartonshire in 2017 she says it felt more urgent than the first time she was elected (in 2005 — she then lost her seat in 2015 and had an “unintended” two years out). “In 2017 I had a different reason for being in politics – I still love my constituency but the issues we’re dealing with are not just quality of life any more, they are ‘How do we run the country by the values that I care about?’” A toy unicorn sits on her desk, a present from her team because, she says in her gentle Glaswegian burr, “it’s a symbol of Scotland but it has also become a symbol of Brexit”.

She noticed a shift just after the referendum. “I was upstairs on the bus going down the Old Kent Road with my son, because when you’re three years old you want to sit on the top deck, and I heard an argument downstairs about space. Then somebody said, ‘Why don’t you just go home?’ Of course, racism went on before people voted for Brexit but it felt like blatant racism had become licensed. That vote was not just about being part of the EU, it was about the kind of country we are — I felt that was being challenged.”

Swinson, who has run three marathons, talks a strong game, but with only 12 Lib Dem MPs in Parliament, her party faces a long road to power. So would she vote for a Conservative majority government Queen’s Speech, or indeed a Labour one? “You look at the Tories and Labour and both have a sea of Brexiteers in leadership roles — that is absolute anathema to the Liberal Democrats, so I would absolutely not [vote with them].”

"I would not support the Tories or Labour — both have a sea of Brexiteers in leadership"

What about forming a coalition? “No. Frankly, look at Corbyn and you have a Brexiteer. That is the only conclusion you can possibly draw from his reluctance to follow what his members and a lot of his MPs have argued for. Both parties are hell bent on delivering Brexit. The next Prime Minister will be a Brexiteer.”

The thought of choosing between Michael Gove and Boris Johnson as the next leader makes her recoil in her black armchair. “It’s not [a choice]. It’s not just that they’re all pro-Brexit, they are all toying with no deal. I can’t understand how we have people who are trying for the highest office in the land, Prime Minister, contemplating that. I do blame Theresa May for a lot of this — she’s legitimised no deal, saying it’s better than a bad deal without recognising that we still have to find a way of dealing with Europe so have to find a deal anyway.”

But without cross-party alliances, do the Lib Dems risk becoming an irrelevant pressure group? “Our job in British politics is clear,” says Swinson. “We are the rallying point for Remain, for people with liberal values. I don’t put any limits on the ambitions of the Lib Dems. Who knows what will happen — no one expected Corbyn to become leader of Labour, no one expected Trump to be President. We have a fracturing of the political system. This is an opportunity. I absolutely recognise the scale of the task but people want a different way of doing politics.” She is working with the former Change UK MPs and “recognises the courage it takes to leave a party you’ve been in for a long time”.

In 2005, aged 25, Swinson was elected MP for East Dunbartonshire, becoming the youngest-ever member of the House of Commons. She still gets called baby of the house at 39, but now she likes it, “given the birthday I’m about to have — no one thinks I’m young except in Westminster so I say I’m ‘politics young’.”

In the coalition government she was junior equalities minister. She’s still friends with Nick Clegg and says he is “a good sounding board, I value his wisdom”. In 2015 her career stalled — she was defeated by the Scottish National Party. Her husband, Duncan Hames, who was also a Lib Dem MP, for Chippenham, lost his seat too. We joke that it was when she left that politics started to go wrong.

“Immediately after I lost my seat people asked if I wanted to stand again,” she says. “I wasn’t ready to make that decision. My husband and I lost our jobs on the same day and we had a 16-month-old baby. We had a lot of practical stuff to think about. I remember my dad being made redundant when I was in my teens and he was in his 50s. It’s bruising and you need space to process it. But I certainly didn’t rule out a return.” She carried on campaigning — she has been doing that since she saw petitions at The Body Shop as a child that encouraged her to write to her MP, and says making a change is what drives her rather than the idea of being Prime Minister. But “when May called the election in 2017 I knew in a heartbeat that I wanted to win back my seat and fight nationalism”.

Throughout her career, Swinson has tried to “take opportunities to push boundaries” when she can. Last year she made history by taking her younger son Gabriel into the Commons with her to listen to a debate. He was 11 months old and had fallen asleep on her. “I didn’t know whether to wake him up and take him back, or miss the speech or just walk in with him. I am a bit of a good girl inside so I thought ‘this is against the rules’. But I was encouraged and the sky didn’t fall in.” She’d already missed a crunch Brexit vote while on maternity leave — and she says Tory chairman Brandon Lewis, who she was paired with, “did the dirty” (he denies it). She campaigned for change and now proxy voting is allowed.

Was bringing Gabriel into Parliament the naughtiest thing she’s ever done? She laughs. “It’s somewhere between Theresa May running through a field of wheat and Rory Stewart smoking opium. I had plenty of fun when I was a teenager.” But at school she was good, “my family had a sense of public service”. Swinson grew up in what’s now her constituency. Her mother is a teacher, who took Swinson into the polling booth with her in the 1987 election (“I told everyone how she had voted the next day, which she didn’t like”). Her father, who died last year from blood cancer, worked in economic development and “always encouraged me to ask questions”. She was in the school debating society and is mortified that in her first debate she was allocated the Tory position and had to argue for the poll tax. The next year she was allowed to choose sides, so she studied the manifestos “and the Lib Dems shone out”.

A 50:50 Parliament tote hangs on her office door and Swinson says while she has never felt “outright sexism”, it is “structural”. “Decisions happen to be made by majority men because the power structures are opaque.” She does note, “overdone chivalry, which only serves to remind you that you are different. One time I snapped because this man was making a point of holding the lobby door for me. I knew he was cheating on his wife; letting me though the door wouldn’t make up for that so I snapped and told him, ‘Just go through the door’.”

There are “more threats to MPs now” and Swinson has a panic button at home. “It’s pretty standard for MPs now,” she says in a matter-of-fact way. “We need a strong, liberal response.”

That brings us back to Brexit. “At the time of the Brexit vote we had Obama. Now the world is much more unstable. There’s the rise of China, Putin, strong men leaders — do you want to be at the mercy of these superpowers? They aren’t going to be giving us great terms on a trade deal; there’s chlorinated chicken, the NHS is on the table. Frankly that is a cause for concern. Or we could have a good deal [staying in the EU] with a bespoke arrangement for the UK with 27 other nations. It’s a no-brainer.” Compounding this is Scottish nationalism. “Brexit is trying to take away my Europeanism,” she laughs. “There is no way you are going to take away my Britishness. I’m Scottish, British, European. I have to fight for that.”

She’s kind about her leadership rival, Ed Davey. “I want to have him alongside me in a superb team.” She pauses. “But I’m the best person to lead this movement.”