In a joint statement, Prime Minister Theresa May and Taoiseach Leo Varadkar called for "new thinking" to break the deadlock between political parties in Northern Ireland.
Negotiations will begin after local council elections in Northern Ireland on May 2, with all the main parties invited.
Ms McKee, a prominent LGBT rights campaigner, was shot dead by dissident Republicans while observing a riot in the Creggan area of Derry.
In their statement, Mrs May and Mr Varadkar said: "We also heard the unmistakable message to all political leaders that people across Northern Ireland want to see a new momentum for political progress.
"We agree that what is now needed is actions and not just words from all of us who are in positions of leadership."
Northern Ireland’s power-sharing institutions have been suspended for more than two years following a row between former governing partners the DUP and Sinn Fein, with disagreement over how to handle the Irish border after Brexit adding to tensions.
Sinn Fein has rejected DUP leader Arlene Foster's latest offer to break the deadlock by reconvening the devolved assembly and executive while continuing talks on difficult issues like same-sex marriage and language rights.
The British and Irish premiers added: "We have agreed to establish a new process of political talks, involving all the main political parties in Northern Ireland, together with the UK and Irish governments, in accordance with the three stranded process.
"The aim of these talks is quickly to re-establish to full operation the democratic institutions of the Belfast/Good Friday Agreement - the Northern Ireland Executive, Assembly and North-South Ministerial Council - so that they can effectively serve all of the people for the future."
Northern Ireland Secretary Karen Bradley and Irish Minister for Foreign Affairs Simon Coveney are expected to hold a press conference later on Friday in Belfast.
The Prime Minister and Taoiseach added: "We understand the complexity of the underlying concerns of all parties, and the need for renewed trust, mutual respect, generosity and new thinking to resolve the issues.
They will review progress at the end of May.
They also agreed that there should be a meeting of the British-Irish Intergovernmental Conference during the same period.
The conference will consider East/West relations, security co-operation, and political stability in Northern Ireland.