Ruth Davidson, leader of the Scottish Conservatives, has increased the pressure on Theresa May to tell the three million EU nationals living in the UK that they can remain after Brexit.

Ms Davidson, who backed Ms May in the Tory leadership election, urged her to move quickly after taking over as Prime Minister to allay fears among the EU nationals that they might be forced to leave the UK. Ms May has refused to make such a promise, saying that residency rights should form part of the negotiations with the EU on Britain’s exit terms – which will also cover the position of the 1.2 million Britons living in other EU countries.

The Scottish Tory leader told Westminster journalists that the status of the EU nationals should be guaranteed. “That's an assurance that I want from the Government, and I want it pretty quick. It’s not enough to say to people who have come here and made a home here, and have made their life here, that we want your labour, that's all we want,” she said.

She added: "We have got to be able also to say we want your brains, we want your culture, we want your passion, we want you in our country, making our country better, and giving these people security."

Ms Davidson dismissed SNP demands for a second referendum on Scottish independence following the Brexit vote as “utterly unjustified” and “completely irresponsible”. She said the SNP should “put the national interest before its own nationalist vision for once”.

But she admitted: “The result is testing this country’s sense of unity. In Scotland, where people voted overwhelmingly to remain in the EU, the result is testing the binds of the Union as well.” However, she pointed out that two million people in Scotland had voted to stay in the UK in the 2014 referendum, more than the 1.6 million Scots who opted to remain in the EU last month.

“The SNP is, as usual, trying to seize on a moment of doubt and uncertainty to declare that the shining star of independence is the answer,” said Ms Davidson. “And it's trying to claim that, as a result of Brexit, there is now a massive groundswell for a second independence referendum. This is simply not true. Even after the EU vote, only four in 10 people in Scotland say there should be an independence referendum.”

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Refusing to join Ms May in ruling out a second referendum on Brexit, Ms Davidson said: ”The honest answer is I don't know what's going to happen – I think it would be highly unlikely.“

She joked about how speedily the Tories had put Mrs May in place, compared with the pace of Labour's attempt to oust Jeremy Corbyn. “Labour is still fumbling with its flies while the Tories are enjoying their post-coital cigarette,” she said.

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Hitting back at Ms Davidson’s speech, Nicola Sturgeon, Scotland's First Minister, said: “It is the selfish and reckless irresponsibility of the Tories that has brought Scotland to the brink of being taken out of EU against our will, with all the damage and upheaval which that would entail for many years to come for our economy, society, rights and place in the world.