The question of whether non-residents of Scotland can vote for the Scottish National party unexpectedly featured in a list of most searched-for terms provided by Google after Thursday night’s leaders’ debate.

Less than a year after the acrimonious independence referendum, the SNP leader, Nicola Sturgeon, clearly impressed voters from around the UK. She came out as the surprise winner of the night in one YouGov survey and scored highly in three other snap polls.

The short answer to the question is no. The nationalist party, whose primary aim is Scottish independence, has only ever fielded candidates in Scotland. (There’s not even an opportunity for anyone in Scotland to vote for Sturgeon herself – she is not standing for a Westminster seat.)

A more nuanced answer is that voters in England and Wales can boost the SNP’s chances of influencing the government by voting Labour, the only party with which Sturgeon could feasibly do a deal.

Sturgeon, who took over as SNP leader and first minister of Scotland from Alex Salmond late last year, has categorically ruled out any pact with the Conservatives, while Nick Clegg has poured scorn on the idea of a rainbow coalition involving the SNP and the Liberal Democrats.

The only credible partner for the SNP therefore is Labour. There are significant differences between the parties on the issues of independence and Trident nuclear weapons, but also some ideological overlap when it comes to the politics of the left.

Sturgeon may have ruled out a formal coalition but both sides have left the door open to the idea of a looser alliance, known as a confidence and supply agreement. This means the smaller party supports the bigger one in key parliamentary votes such as the budget, and acts in other votes on a case-by-case basis.

In the debate, Sturgeon clearly signalled she was open to working with Ed Miliband, at one point saying: “I agree with Ed.” She challenged the Labour leader to join her in seeking an end austerity and said the SNP was his “ally” in trying to roll back a tide of privatisation in the NHS.

Her decision to open the debate by making an appeal to the people of the other UK countries was significant. “I know it’s not just people in Scotland who feel let down by Westminster politics. My message to the people of England, Wales and Scotland is one of friendship,” she said.

This matters because the Conservatives have assumed people in England fear the prospect of an alliance between Labour and the SNP, running attack ads showing Miliband in the pocket of Salmond.

This is certainly true for some voters, illustrated by Farage’s willingness to say the English are “cheesed off” by the amount of money flowing over Hadrian’s wall. But the polls giving credit to Sturgeon suggest this is not a view shared by all.

The dynamics are further complicated by the fact that a good performance by Sturgeon is not good news for Scottish Labour, which faces losing dozens of seats.

This week a ComRes poll for ITV News focusing on Labour’s 40 Scottish seats found that the SNP had a six-point lead, putting Sturgeon’s party on the brink of winning about 28 new seats and close to becoming the third largest party at Westminster.

While Sturgeon indicated a willingness to work with Miliband, he needed to prove that her party was not relevant or necessary in order to persuade Scottish voters to return to the Labour fold. That is why the Conservatives appeared to enjoy the Miliband-Sturgeon exchanges so much and were talking up her performance.

Michael Gove, the Tory chief whip, described her as the “most impressive debutante” in the debate, before going on to warn of the perils of a Labour-SNP vote. This is part of what Labour has described as the “unholy alliance” between the SNP and Conservatives where both think they benefit from talking up the chances of Sturgeon’s success.

The key for the SNP will be to emerge victorious in Scotland while making sure voters in the rest of the UK are not sufficiently spooked by this prospect to opt for the Conservatives over Labour.