Sinn Féin has refused to nominate a new deputy first minister in Northern Ireland, triggering the collapse of the power-sharing government and fresh elections.

Theresa May earlier held talks with the first minister, Arlene Foster, and Martin McGuinness, who resigned as deputy first minister last week, in a late effort to save the Stormont government, stressing that it had a role to play in Brexit negotiations.

May’s spokeswoman said: “She spoke to both Arlene Foster and Martin McGuinness to encourage them to use what time was left today to try to find a resolution to the situation in Northern Ireland, outlining the importance of finding a way forward – particularly in the context of wanting to make sure that Northern Ireland has a voice in the UK’s exit from the European Union as we approach the critical period before triggering article 50.”

A short time later, Foster said at Stormont: “Northern Ireland does not need an election, it needs stable government.” She said Sinn Féin had “forced an election that risks Northern Ireland’s future and stability and which suits nobody but themselves”.

McGuinness, who is being treated for a rare condition that attacks the heart and other organs, resigned last Monday in protest at Foster’s refusal to step aside while a public inquiry is held into an energy scheme whose costs have spiralled out of control.

If Sinn Féin fails to nominate a new deputy first minister by 5pm on Monday, the UK’s Northern Ireland secretary, James Brokenshire, will have to set a date for fresh elections. Under the complex rules of power-sharing in Northern Ireland, a government cannot survive if one section of the community refuses to participate.

McGuinness tweeted: “In conversations this morn with the British PM & her sec. of state I said society & I felt badly let down by both [Foster’s] DUP & the British govt.”

It is unclear whether McGuinness will stand as a candidate in the forthcoming elections. Sinn Féin is keen for his name to be on the party’s slate in the city of Derry, but it is understood his family is concerned about the impact of an election campaign on his health.

The main parties represented in the devolved parliament have already started electioneering even before the official start of the campaign. Over the weekend Foster attended a party rally at a hall belonging to the Orange Order in her native Fermanagh, and Sinn Féin revealed its slogan for the contest – “Get even”.

At a social club run for the benefit of ex-IRA prisoners in west Belfast, the Sinn Féin finance minister, Máirtín Ó Muilleoir, said: “People are looking for an opportunity to say they have zero tolerance when it comes to corruption, zero tolerance to bigotry, and that they want respect and equality. That is what will be on Sinn Féin’s ballot paper.”

The cross-community Alliance party said it wanted the campaign to focus on allegations of corruption and cronyism associated with the renewable heating incentive scheme as well as other scandals.

Naomi Long, the Alliance leader, said: ”Our party isn’t to blame for the snap election called as a result of the DUP’s and Sinn Féin’s mess. But, unlike some, we are not running scared of an election. We will be facing the people confident, and offering the electorate an alternative to the secrecy, corruption and cronyism that they are sick of at Stormont.”

On Sunday Brokenshire had said no alternatives to power-sharing were being contemplated, including direct rule or joint authority between London and Dublin.

However, given the bad blood between the DUP and Sinn Féin, negotiations to form a post-election government are expected to be long and tortuous, with no guarantee that the parties will agree to form a coalition.

The hardline unionist assembly member Jim Allister predicted that efforts to build a new coalition could fail. “Those who have chosen to believe the lie that Sinn Féin wants to make Northern Ireland work have had a rude awakening over the past few days,” he said. “Sinn Féin will be seeking further concessions after the election, when there is nothing more for unionists to give.”