Prime Minister Rishi Sunak and Taoiseach Leo Varadkar need to co-chair a summit meeting to chart a course back to devolution at Stormont, Sinn Fein vice president Michelle O’Neill has said.

The republican party insisted its success in the local government elections was a “monumental endorsement” of its call for the powersharing institutions to be restored.

However, DUP leader Sir Jeffrey Donaldson said he now has a renewed mandate from the council elections to go to the UK Government and seek solutions on the Northern Ireland Protocol.

The elections to 11 local councils saw Sinn Fein emerge at the weekend as the largest party in local government for the first time, with 144 seats, a rise of 39 from 2019.

The party replicated its result in the Assembly election last year when it became the largest party at Stormont.

The DUP reinforced its position as the dominant force in unionism by winning 122 seats, the same number as in 2019.

The poll took place against the backdrop of the DUP’s decision to collapse the Stormont powersharing institutions in protest at post-Brexit trading arrangements.

The DUP is seeking further steps from the Government to ease its trading and political concerns about the protocol before it commits to a Stormont return.

Sinn Fein president and vice president Mary Lou McDonald and Michelle O’Neill appeared together at a Belfast hotel.

Mrs McDonald said her party had “broken new ground”.

She said: “Thursday was the people’s day and they have now spoken loud and clear.

“The result has been a monumental endorsement for Sinn Fein’s positive and progressive platform to restore government (at Stormont), to invest in the health service, to support people through the cost-of-living crisis and to deliver first-class council services.”

Mrs McDonald said a significant motivating factor in the election was public frustration that Michelle O’Neill had been prevented from becoming first minister at Stormont due to the DUP veto on powersharing.

“Even beyond nationalism and republicanism there is a recognition that Michelle O’Neill should now be serving as a first minister for all,” she said.

Ms O’Neill reiterated her call for an urgent meeting of the British-Irish Intergovernmental Conference.

“It should be at the highest level, at taoiseach and prime minister level, and we want to see a plan to have the institutions back up and running,” she told reporters in Belfast.

Ms O’Neill said the DUP’s ongoing blockade on Stormont was “totally unacceptable”.

“There’s no doubt the political landscape across this island is changing, but Sinn Fein is leading on that change and we are in this decade of opportunity and people want new beginnings,” she said.

“We said that we will work for all, and that I will be a first minister for all, and I’m ready to start that work today, I’ve been ready to start that work since last May.

“One party’s boycott of the Assembly cannot go on and the Executive must be formed now.

“So, it’s now time to make politics work and deliver for people right across this island.

“That’s what the public have just endorsed in this recent election again.”

Ms O’Neill said it would not be acceptable for a return to Stormont to be delayed until the autumn.

Appearing at Stormont where he was flanked by a number of his DUPs and MLAs, Sir Jeffrey said he had delivered a very strong mandate.

He added: “That means the DUP has a renewed mandate to go back to the Government and seek the solutions that we need on the Northern Ireland Protocol, to restore our place within the United Kingdom and our ability to trade with the rest of the UK.”

He added: “I believe the mandate we have been given in the council is a mandate to finish the job.

“That means getting Northern Ireland’s place within the United Kingdom restored, protected in law and getting Stormont back up and running and delivering for the people of Northern Ireland.”

Sir Jeffrey said his party was committed to seeing Stormont restored, but it had to be on “solid foundations”.

He said: “We’re committed to seeing this place (Stormont) properly restored and functioning and delivering.

“Whether that is more effective, more efficient public services, whether that’s tackling the problems with our public finances.

“We will be at Westminster making the case to the Treasury and we will be here preparing for the restoration of the Assembly – but on a solid foundation.

“That was a key message that we put forward on our election campaign.

“This place, Stormont, needs to be restored on solid foundations.”

He added: “Progress in Northern Ireland is only made with the support of unionists and nationalists.

“Whatever changes have taken place during the election, that reality remains a constant.”

Alliance party leader Naomi Long said the lower voter turnout in some predominantly unionist areas has an impact on the mandate of the DUP leader to continue to boycott Stormont.

The cross-community Alliance party is now the third largest party in local government after gaining 14 seats in the election, giving them a total of 67 councillors.

“There is absolutely nothing productive coming from holding back government in Stormont whatsoever,” she said.

“And if Jeffrey really wants to look carefully at what mandate, and what support the party has, he should look at the number of voters who stayed at home, unionists voters who felt that there was nothing worth coming out to vote for.”

She added: “I think it should be a concern to all of us when so many unionists chose to stay at home, that is of concern in politics.

“If Jeffrey doesn’t see that as a concern, then I don’t see unionism changing its course, and I don’t see unionism faring well into the future.”

UUP leader Doug Beattie said Sinn Fein’s electoral success was a failing of the unionist parties.

The UUP lost 21 seats in local government, bringing them to 54 councillors.

“Nationalism and particularly Sinn Fein have had a fantastic election, you can’t take that away from them. They had the resources, they had the people. They had a good plan and they are now the largest party in local government and that’s our failing,” he said

“They haven’t grown that much right across nationalism, but unionism has shrunk and that’s the problem that we have to look at.”

He added: “I think unionism across the board, whether it’s the DUP, the Ulster Unionist Party, the TUV or any other unionist party needs to understand that positivity, confidence is what’s going to inspire people to get up in the morning and vote.

“But more than that is having a vision for the future because people want to know what we want to have Northern Ireland looking like in five, 10, 15, 20 years from now, not just when an election is around the corner, but right into the future. That’s what we need to focus on.”

SDLP MLA and leader of the opposition at Stormont, Matthew O’Toole said the DUP using its mandate to continue to boycott Stormont was “the most strategically stupid thing the leader of unionism could possibly do”.

The SDLP won 39 seats in the local elections, a loss of 20 from their 2019 result.

“I think it’s clear now that the DUP insisting upon using a mandate to boycott at the expense of basic governance here and the expense of public services, is the most strategically stupid thing the leader of unionism could possibly do,” he said.

“I don’t know how anyone who leads unionism could look at the election results last week and think this is good for the union, this is good for the constitutional status quo.

“So if Jeffrey Donaldson doesn’t wise up, lead his party, and frankly, lead unionism back into this place (Stormont), the government – where we will be a constructive opposition – then there’s not much I can do to help him, completely extraordinary.”

Downing Street said it remained hopeful parties could return to powersharing.

A No 10 spokesman said: “It remains our position that we’re hopeful the parties will come together and see the Assembly and the Executive return to work.

“We have consistently said that stable, effective and accountable government in Northern Ireland is in the interests of everybody.”