Political leaders in Northern Ireland have begun talks in a fresh effort to break the protracted impasse on power-sharing, with the prime minister, Theresa May, and the taoiseach, Leo Varadkar, pledging to review progress at the end of the month.

The talks are aimed at restoring devolved government, which collapsed spectacularly in January 2017 after a bitter row between the Democratic Unionist party and Sinn Féin and a breakdown in trust.

The UK and Irish governments set out a road map on Tuesday for the talks with a pledge to hold a weekly evaluation by the Northern Ireland secretary, Karen Bradley, and Ireland’s deputy prime minister, Simon Coveney.

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The region has been left without a working government for for than two years but renewed efforts to get Stormont up and running again have been made after the killing of the journalist Lyra McKee by dissident republican terrorists.

Arriving at Stormont House in Belfast for the opening meeting on Tuesday, Sinn Féin’s president, Mary Lou McDonald, said her party was there to “do the business”.

She has taken over at the head of the party since the retirement of Gerry Adams and the death of Martin McGuinness a few months after the collapse of power-sharing.

While Sinn Féin and the DUP appeared not to be punished in last week’s local elections over the failure to compromise, politicians are facing mounting public pressure to find consensus amid concerns that violent extremists are exploiting the power vacuum.

Many have voiced scepticism that talks will result in a deal by the end of May, but there is hope the establishment of the process will engender a positive environment that will eventually lead to success.

May and Varadkar will review progress and consider any necessary further steps at the end of the month, such as the continuation of talks, according to a statement by both governments.

The involvement of all five main parties in Northern Ireland has the potential to inject some consensus-building around the table, missing from previous talks between Sinn Féin and the DUP.

Colum Eastwood, the leader of the Social Democratic and Labour party (SDLP), cautioned that the new talks could not be “privatised” into bilateral negotiations confined to the two largest parties.

“I think the mistake has been made over the last couple of years that this process has been privatised between the DUP and Sinn Féin,” he said. “The strategies that have been employed by those two parties up to now haven’t worked, so I think it’s time for a rethink.”

McDonald said the stalemate was “not acceptable” or “sustainable” and she believed there was a path to success if everyone showed leadership. “If everybody is prepared to respect the clear public desire for equality and people’s rights to be recognised and delivered on, we can find our way back to power-sharing,” she said.

Arriving at the talks venue, the Ulster Unionist party leader, Robin Swann, said the process must not be just “window dressing”.

“What we need to actually get to this set of talks is to sort out a governance structure that doesn’t give a veto to one party that can simply crash democracy in Northern Ireland because of their will,” he said.

“If this is simply five parties sitting round a table again to re-establish red lines, we’ve let the people of Northern Ireland down, and if those parties who come in with red lines established are sticking by them, then they are letting the people of Northern Ireland down.”

The DUP and Alliance party, the other local participants in the meeting, did not speak to the media outside Stormont House prior to the meeting.