Ms O’Neill said there could be no delay to the restoration of the Stormont powersharing Executive and her nomination as first minister, following her party’s election victory.
But DUP leader Sir Jeffrey Donaldson has stated that he will not re-enter the Executive without “decisive action” from the UK Government on the Northern Ireland Protocol.
This means that prospects of any quick return of the devolved powersharing Executive at Stormont are diminishing.
MLAs returned to Parliament Buildings on Monday and party leaders were also holding separate meetings with Secretary of State Brandon Lewis.
It came as Prime Minister Boris Johnson said he does not plan to be personally involved in the Northern Ireland talks and Downing Street played down reports of a Cabinet rift over the protocol.
Brinkmanship will not be tolerated where the north of Ireland becomes collateral damage in a game of chicken with the European Commission
Thursday’s historic election saw Sinn Fein become Stormont’s largest party.
Speaking to the media on Monday, Ms O’Neill said: “The people have spoken and they have spoken very clearly.
“The message is one of hope, it is also one of optimism for the future, for the political leaders to work together and to make politics work.
“That is my commitment as a political leader and as an incoming first minister.
“The electorate also demands that the parties get back down to business, to elect a speaker, to sit in the Assembly, to have it function, to appoint a first minister and a deputy first minister, to form a new Executive.”
She added: “Today we have met with the British Secretary of State, have spoken with the Taoiseach and will be engaging with other party leaders and my message is clear.
“As democrats, the DUP, but also the British Government, must accept and respect the democratic outcome of this election.
“Brinkmanship will not be tolerated where the north of Ireland becomes collateral damage in a game of chicken with the European Commission.
“Responsibility for finding solutions to the protocol lie with Boris Johnson and the EU.
“But make no mistake, we and our business community here will not be held to ransom.”
Speaking following his meeting with Mr Lewis in Belfast, Sir Jeffrey said: “We have had a meeting this morning with the Secretary of State and we have made our position clear to him.
“It is the position we have held before the election, throughout the election campaign and will continue to hold, and that is until we get decisive action taken by the UK Government on the protocol we will not be nominating ministers to the Executive.
“We want to see stable political institutions, we want to be part of the Executive, we want to play our part and fulfil the mandate we were given by the people of Northern Ireland.
“We are also clear given the damage and harmful impact the protocol continues to have on Northern Ireland, driving up the cost of living, harming our economy, impeding the ability of businesses to trade with our biggest market and fundamentally undermining political stability, undermining the principle of consensus politics.”
The DUP leader added: “We need this to be resolved.
“When we re-entered the Executive back at the beginning of 2020 we did so on the basis of a clear commitment by the UK Government under the New Decade, New Approach agreement to address and protect our ability to trade with the rest of the United Kingdom.”
The centre-ground Alliance Party made significant gains in the election, increasing its seat count by nine to 17, becoming the third largest party at Stormont behind Sinn Fein and the DUP.
Leading her enlarged team into Stormont on Monday morning, Ms Long told the media they were turning up for work as they had promised voters during their election campaign.
“I want us to sit down, get the negotiations under way on the programme for government and the budget, and I want to see us getting government up and running as quickly as possible,” she said.
“We’re going to see a functioning Assembly hopefully pretty soon, that will be the easier part of all of this, but to get a functioning Executive we need the DUP to step up to the plate.
“With power comes responsibility, and people now need to take the responsibility seriously.
“I think the DUP need to reflect on their mishandling of Brexit, on the fact they had huge influence at Westminster and overplayed their hand.
“It would be foolhardy for them to overplay their hand with devolution because that is a gamble that Northern Ireland can’t afford to take.”
Ms Long was also critical of statements by government ministers that the Northern Ireland Protocol is threatening the Good Friday Agreement and preventing formation of government at Stormont.
“What is threatening the Good Friday Agreement is the British Government not playing an open hand with the people of Northern Ireland, using and abusing the protocol as an issue to try to garner support for their own cause,” she said.
SDLP leader Colum Eastwood urged the DUP to nominate ministers for a new executive.
He said: “We know who got the mandates, we will support them when we have to, we will also hold them to account, but they should get on with it today and form a government.”
The Stormont Executive has been unable to fully function since February when First Minister Paul Givan resigned as part of the DUP’s efforts to force action against the Northern Ireland Protocol.
Sinn Fein’s election victory means that their Stormont leader, Ms O’Neill, should be in line to become the first nationalist or republican First Minister.
However the DUP, as the second largest party, must nominate a deputy first minister to serve alongside her in the joint office.
The Northern Ireland party leaders are due to meet at Parliament Buildings later in the afternoon.
Meanwhile, the Prime Minister’s official spokesman said that people in Northern Ireland “deserve a stable and accountable devolved government”.
No decisions have yet been taken on the way forward. The Deputy Prime Minister made clear the situation, it’s very serious
On whether there is any plan for the Prime Minister to get personally involved in the talks or involved in the talks with the EU, he said: “No plans for that.
“This has been led by Brandon Lewis.”
When asked about the Government’s position on the protocol and whether there are divisions within Cabinet over proposals to unilaterally scrap it, the Prime Minister’s official spokesman said: “I wouldn’t say that at all.
“I think our preference has always been for a negotiated solution to fix the protocol and we have been clear that we will take further steps if solutions can’t be found.
“No decisions have yet been taken on the way forward.
“The Deputy Prime Minister made clear the situation, it’s very serious.”