Ruth Davidson has had a good summer. At the age of 38, she has finally bought her first house. It’s a two-bedroom mid-terrace in an Edinburgh suburb that she will share with her Irish fiancée, Jen, and their cocker spaniel, a failed gun dog called Wilson (“It’s just as well he’s handsome because, by God, he’s stupid,” she tells me). The hyperactive leader of the Scottish Conservatives is eager to put down roots. “I’ve always moved for work,” she says. “I worked out that since I left home to go to uni at 17, I’d had more than 20 flats. This is the first time I’ve had a home. It’s nice.”

On 29 August, her opposite number in the Labour Party, the well-liked Kezia Dugdale, resigned. Her replacement is likely to drag the Scottish party in a more Corbynite direction on issues such as nationalisation, taxation and public spending. This will put pressure on the SNP – now the party of choice for many disaffected Labour lefties – to do the same. That would leave space in the centre ground that Davidson’s Tories will be more than happy to fill.

“If I’m perfectly honest, I am by nature a centrist,” she says. “I’m fairly hard-core on some justice and fiscal policies. I’m a proper Tory there. But in terms of social policy and things like that, I’m absolutely a centrist. But it’s because I think it’s right. It brings people with you and, if you’re looking towards [forming a] government in a way that as a party in Scotland, five to ten years ago, we could never have conceived, it’s about bringing people with you and making the arguments for being bold and radical.”

This sounds familiar. Is the great young hope of British Conservatism a much more youthful, female version of Tony Blair? That won’t go down well in the Shires or the leader columns of the Daily Mail. “No! I didn’t go to Fettes, I don’t own… rental properties around the world, I don’t holiday with pop stars, so I don’t consider myself to be a Tory Tony Blair. There’s some things I think he did very well. I think in terms of foreign policy, his idea of humanitarian interventionism that he used in Sierra Leone and in Kosovo was bang on. It was the right thing to do and it saved lives. However, I’m probably the only Tory leader who has been on one protest march in their life and that was against the Iraq War in 2003, so there are things I don’t agree with him on. Actually, I joined the Territorial Army about a month later because I wanted to serve in some way – though not in Iraq.”

Ruth Elizabeth Davidson grew up in a Presbyterian family in Selkirk, where her father worked in a wool mill and she attended a comprehensive school. After a career in broadcast journalism, she entered politics and became leader of the Scottish Tories in 2011; she has since revitalised the party in one of the great contemporary political feats. With Davidson at the helm, a party that was wiped out in the 1997 election (it won none of Scotland’s 72 Westminster seats) and that had shown only a flicker of life since then has supplanted Labour as the official opposition at Holyrood. In June’s general election, the Tories won 13 seats (out of 59) in Scotland, an increase of 12. Between the 2015 and 2017 general elections, the Scottish Tories put on more than 320,000 votes; in the May local elections, they more than doubled their share of Scottish council seats to 276.

There is a good chance that in 2021, when the next Holyrood elections are held, Davidson will find herself leading Scotland’s largest party and becoming first minister. Already she regularly attends Theresa May’s political cabinet in London and is spoken of at Westminster as a future prime minister – some would parachute her into No 10 tomorrow if they could. Members of her small back-room team say that they are besieged by media interview requests and invitations from around the world. Everyone wants a piece of Ruth Davidson’s magic.

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When we meet in her small office on the Conservative floor of the Scottish Parliament, I sense the low hum of military-style planning, even though Holyrood is still in recess. After ten days in Ireland, Davidson is rested and recharged. “I think along with almost every other person involved in politics [or] journalism about politics, and the voters, I went into the summer absolutely knackered. But I’m ready to go again. We’ve had a really good 18 months. We’ve had three elections where we’ve come from third to second each time, we’ve more than doubled our number of MSPs, more than doubled our number of councillors. We’ve gone 13 times our number of MPs, though that maybe talks more about the base level than anything else…”

It’s certainly true that the old joke about there being more giant pandas in Scotland (there are two) than Tory MPs (there was one) has run its course. “The pandas are going to have to do a lot of listening to Barry White music to catch up with us now.”

Yet Davidson is far from satisfied. “I don’t want this to have been a peak. This is a platform for us to build on. In the five-and-a-half years I’ve been leader, between referendums and elections, I’ve fought eight national campaigns. Scotland is tired of politicians shouting at each other with no end product, and we need to use this period – which is the first we’ve had in years with no imminent election – to reduce the temperature in Scotland and in the political discourse. We need to use it to do some of the heavy intellectual lifting that’s not been done in this place [Holyrood]. We need to start asking questions about long-term solutions in important policy areas.”

The 20th anniversary of the referendum that licensed Tony Blair’s creation of the Scottish Parliament falls on 11 September and is inevitably inspiring some reflection and soul-searching north of the border. Not many would claim that the institution’s first two decades have been a shining example of policy innovation and political daring. “Are we as a country more dynamic, braver, more advanced, better educated, with better health than 20 years ago? I’m not so sure,” says Davidson. “Honestly, I think it’s been timid. I think devolution was designed to be more ambitious than what previously existed, and I’m not sure that ambition has been realised within this building at Holyrood.”

If given the opportunity, she wants to make good on the parliament’s potential. She accuses her SNP rivals of big talk but little action: “They’ve been very good at saying whatever issue of the week they’re getting hammered on is their top priority and that they’re going to have a commission, or there’s going to be a review. At some point, you actually have to start making tough decisions.”

The day after our interview, Davidson unveiled proposals for a series of new towns in Scotland and for 25,000 homes to be built annually. On education, she wants to encourage innovation by giving head teachers autonomy over budgets. She aspires to boost the status of the teaching profession, allow high- and low-performing schools in the same localities to “buddy up”, and encourage different types of school to open, including technical and state-funded schools that opt out of local authority control.

Davidson wants to introduce Teach First, which fast-tracks high-performing graduates into the teaching profession, to Scotland. “We used to pride ourselves at being the best in the world at education. Well, let’s have a bit of humility and let’s look at what’s happening in the world that’s better than what we’re doing.

“I understand that the SNP were trying to keep a broad collective together because they were working towards the goal of independence, but it’s not good enough that an entire generation’s life chances have been thwarted because you’ve been afraid to take on the teaching unions, or you’ve been afraid to make the changes that perhaps parents wouldn’t understand and you’d have to explain to them.”

Measures to tie the NHS and social care together will receive proper attention in the next few years, she says, as will the economy. “Part of centrism is about understanding the need for private industry, private enterprise, free trade, the idea that you can lift all boats. Inequality in the UK is at its lowest level for 36 years, but it doesn’t feel like that to people out there. They see these millionaire footballers or Russian oligarchs in London with their gold-plated Bentleys while they’re struggling and that disconnect is really tough.”

The ambition is clear, although the dissimulation and cant of the conventional political interview are replaced by a refreshing frankness. “We’re getting ready to change from a strong opposition to looking like an alternative government of Scotland,” she says. “We don’t look like that now. We know that. We’ve got a lot of work to do, but we’re up for it. I have to make sure I’ve got the team, the vision, the policies, the ideas, and that we’ve got the tone right – the civility that we can bring back into politics in Scotland, because it’s been at fever pitch for a really long time.”

She continues: “We have people who are serious, thoughtful, who probably ten years ago wouldn’t have changed career to do politics. But this big, cataclysmic referendum [in 2014] happened where people said, ‘The Scotland I want is worth fighting for.’ Whether you were for Yes or No, it dominated so much that a lot of people who would have just sat on the sofa and shouted at Question Time decided to get off their backsides and do something about it.”

In Scotland’s predominantly leftist political culture, there are those for whom a Tory – centrist or otherwise – can never be anything more than a stone-hearted friend of the moneyed elites. Davidson’s electoral success and personal popularity are all the more luminous when contrasted with the miscalculations and missteps that have gored the reputations of several senior London colleagues, including the Prime Minister and the Foreign Secretary.

Davidson says she isn’t worried about cross-contamination, but an indication of how Westminster decisions can trip her up came earlier this year when the UK government announced plans to restrict child tax credits to the first two children. An exemption was announced for women whose third child was a result of rape, but campaigners were furious that victims would be expected to prove their circumstances to the DWP.

Davidson defended the so-called “rape clause” and found herself in a difficult spot. “It was said I looked uncomfortable talking about it – well, yes. But do I want to make sure people who have had children in the very worst circumstances have the financial support that they need? Yes, I do. Nobody was putting forward a better way of doing this.”

Were her opponents in Scotland using the issue to tarnish her reputation? “Look, I’m not going to say that. But it’s interesting that even Jeremy Corbyn didn’t think it was an issue on the campaign trail.”

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Davidson was a staunch Remainer. She aggressively debated Tory Leaver colleagues during the referendum campaign – most notably roughing up Boris Johnson, for whom she has little time, at a debate at Wembley Arena in London. She accepts that Brexit “is going to happen. You’ve got no major political party likely to be in government advocating that it doesn’t happen and no electoral event that would give them the mandate to stop it before it happens.”

Yet she is far from uncritical of the government’s performance. Of the fraught beginning to the Brexit negotiations, she says, “I think one of the things the UK government didn’t do that they should have done was pitch-roll this: remind the British public that when it comes to European negotiations – and we’ve had several decades of them – we are told no until five past midnight and then suddenly a deal gets done in the wee small hours of the morning. I don’t think the country was prepared for this period that we’re currently in. People in a room talking and then walking out and up to a bank of microphones and saying entirely different things while standing next to each other is part of what negotiation is. I think the UK government has not just an obligation but a duty to negotiate as hard as they can on behalf of the country.”

What is her biggest concern about the impact of Brexit? She pauses. “Interesting question… My real fear is that if there’s a short-term economic hit, we don’t bounce back from it.”

Would she like a prolonged transition period during which Britain maintains access to the single market? “I’m for free trade and want to make sure that people from Scotland and the UK have access to – and the greatest ability to operate within – the single market, which I believe are the exact words the Prime Minister used in her Lancaster House speech back in January. The mechanism for how we get to that I’m less aerated about, as long as that’s where we get to.”

We have reached, at last, a mention of the invisible Prime Minister, in office but not in power, counting down the days until her colleagues decide to free her from the burden of empty leadership. I say that it’s brave of Theresa May to get on with the job each day. It can’t be fun. “She’s absolutely straight down the line,” Davidson says. “She’s not a game player. And the kind of clichés that you hear about her, about her believing in service and public duty, are absolutely true. Everything that she said about being there for the long haul, as long as the party and the country want her – she will get up and she will put in a shift.”

Could Davidson end up occupying that hottest of seats? David Cameron once told me that he “never put a limit on her abilities and ambitions. She has got what it takes in politics. She’s got oomph, she’s got spirit, she’s got brains.”

One friend who has watched her astonishing progress concedes that even Davidson has been surprised by her success. “She has had to get her head around how good she is and how much potential she has – that she can play on the biggest of stages. Each time we think she’s reached a plateau, she climbs the next one. I genuinely think she could do just about anything she wants to, and maybe she’s starting to believe that.”

For Ruth Davidson, the next plateau is in sight. “When 2021 comes around, people will be looking for a first minister, and the option they’re going to have is Nicola Sturgeon again or me,” she says. It’s a remarkable statement, given recent history, to come from the lips of a Scottish Tory leader – but she means it, and we should take her seriously.