An initial round of talks between May and DUP leader Arlene Foster ended with no agreement on Tuesday, although both sides said they were hopeful of a deal.
May does not necessarily need a firm deal from the DUP before opening parliament and might hope that she would receive the necessary backing anyway.
Even without the impact of the London fire, the talks were said to be making "slower progress".
"Look at what the Tories (Conservatives) have managed to do to the United Kingdom in the space of just one year, firstly calling a divisive and reckless European Union referendum. then having lost that gamble pursuing a hard Brexit path", Scotland's first minister, Nicola Sturgeon, said.
After House Speaker John Bercow was re-elected without challenge, a chastened May quipped: "At least someone got a landslide".
May has dismissed calls to resign following the dismal election result after calling a vote three years early in the hope of bolstering her slim majority ahead of the Brexit talks.
"However, while talks are ongoing it is important the Government gets on with its business and we are confident there will be sufficient support across the House for passing the Queen's Speech".
"It will take us several months to draw out the conditions of an orderly withdrawal.so let's not waste time", he said.
The talks are being closely watched in European capitals as they could delay the expected start of Brexit negotiations next week.
Labour leader Jeremy Corbyn countered with a bit of previously unforeseen swagger, wearing a huge red rose - his party's symbol - in his lapel as he sparred with May.
"Westminster has brought us austerity, has brought us hardship, and it has hurt the working class people in our communities", one of their MPs told reporters at a news conference in London. And - while they may be rescheduled - Brexit talks are technically meant to begin in just five days' time. She lost her majority and now needs to broker a deal with the DUP.
"The parliamentary arithmetic is such that we are going to have to work with everyone", he said.
Sir John suggested that any agreement which involved large increases in public funding for Northern Ireland in return for the DUP's support could create resentment among voters in other parts of the UK.
"The two issues - Northern Ireland and Brexit - might end up making the other more hard, in a vicious circle", Usherwood said.
The Democratic Unionist Party controls the Northern Irish Assembly.
Foster's rivals in Northern Ireland, such as Sinn Fein's Gerry Adams, have objected.
"It's imperative that both governments recommit to the word, spirit and implementation of the Good Friday Agreement if there is to be any prospect of re-establishing the Executive".
The stakes for May are high.
The Conservative source said: "We're confident of getting an agreement, we're confident that the Queen's speech will be passed".
Labour's unexpectedly strong second-place showing has thrown national politics into disarray.
As European leaders tried to fathom exactly how Britain would begin the negotiations, German Finance Minister Wolfgang Schaeuble said he wanted a Brexit deal that would limit negative consequences for the bloc but nor did it want to weaken Britain.