Prime Minister Theresa May is in the fight for her political life after last week's disappointing showing in the United Kingdom election, with no shortage of people and factions to appease.

The Queen's speech, in which the government sets out its program for the next parliamentary session, was set to take place on June 19. But there are questions if that date is realistic because there are key substantive discussions to be held both within her Conservative Party and with its rivals and would-be partners.

With a fall from 331 to 318 seats in the House of Commons, May is first tapping the socially conservative Democratic Unionists (DUP) of Northern Ireland (DUP), winners of 10 seats, for an arrangement to keep her minority government in power.

DUP leader Arlene Foster is due to arrive in London to continue negotiations with May on Tuesday.

Foster is arguably foremost among those seeking to press leverage over a humbled May, but there are others.

Ruth Davidson

Davidson doesn't have a seat in the House of Commons but as leader of the Scottish Conservatives, she could be a key player in the internal wrangling over how May can piece together a productive government.

The Conservatives are only in a minority government position at all because the Scottish Tories gained at the expense of Scotland's National Party.

Scotland's Conservatives finished with 13 seats, their highest seat total in an U.K. election since 1983, and with their highest popular vote share since 1979. That bump was presaged by gains in last year's Scottish parliamentary elections and local elections in March.

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Ruth Davidson, the leader of the Scottish Conservatives, leaves a cabinet meeting at Downing Street in London. Scotland voted to remain in the European Union last year, while Davidson bristles at the DUP's stance on social issues. (Hannah McKay/Reuters)

Scotland voted overwhelmingly to Remain in the 2016 European Union referendum. As a result of May's poor showing last week, Davidson said that May's wish for a hard Brexit is off the table.

Davidson, who met May on Monday, wants "a shift in thinking" on Brexit, with more emphasis on the economy and less on reducing immigration numbers, a source close to her told Reuters.

Davidson will also be adamant that any agreement with DUP, which opposes gay marriage, stays away from rolling back LGBT rights and same-sex marriage. Davidson is gay and engaged to her partner.

Leo Varadkar

Varadkar has taken over as leader of Ireland's Fine Gael and in will soon be sworn in as the country's leader.

Varadkar's predecessor, Enda Kenny, said earlier this year Ireland would "remain an enthusiastic member of the EU." The country's small economy has rebounded since the global economic crash of 2008 and wants the benefits of being integrated in the single market

There are also the considerations closer to home. Ireland does not want a "hard border" with Northern Ireland and in April struck a deal with the EU in which Northern Ireland could remain part of the union if it left the U.K. to become part of an integrated Ireland.

The Remain side in Northern Ireland won with nearly 56 per cent of the tally although there was disparity geographically, with the south and west being much more opposed to Brexit.

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Leo Varadkar, the presumptive next leader of Ireland, wants to retain stronger ties with the EU and preserve peaceful and open relations with Northern Ireland. (Aidan Crawley/EPA)

Ireland's violent recent history from the 1970s and 1990s is also a factor, at least symbolically.

Britain is a co-guarantor of the Good Friday Agreement in 1999 that ushered in a new era of relations and both Varadkar and Sinn Fein leader Gerry Adams have made noises that teaming up with the DUP would go against the terms of the agreement for Britain to be an impartial broker in Northern Ireland.

Varadkar spoke on Monday and is insistent that the British government of the day, whichever party it is, will "not to be too close to any particular party in [Northern Ireland], whether it's a nationalist or republican party or whether it's a unionist party."

Boris Johnson

May made her pitch on Monday to members of her party to remain as leader.

George Osborne, the longtime MP and recent chancellor of the exchequer who did not run in this election, called May a "dead woman walking" on Sunday, but that characterization was rejected by May ally David Davis, the Brexit secretary.

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May sits next to Foreign Secretary Boris Johnson on Monday. Johnson said publicly the party should rally around May. (Leon Neal/Reuters)

The most high-profile challenger who could succeed May is Boris Johnson, the former London mayor and current foreign secretary. Johnson is decisive and outspoken but has been tagged as "gaffe-prone."

At least publicly, Johnson was calling upon colleagues to rally around May.

There is a recent example for May of an international leader in a similar position moving forward productively after a hung Parliament.

Labour Leader Julia Gillard in Australia was able to forge a working relationship with MPs from other parties in 2010. The arrangement lasted three years but former PM Kevin Rudd emerged within her party to wrest the leadership back.

In contrast to the Australian situation, where the economic forecast was positive, the United Kingdom is awash in uncertainty after two general elections, a leadership change and the vote to leave the EU in the past two years.

The bottom hasn't fallen out on the British economy in the year since the Brexit vote, but inflation is rising and all economic growth forecasts are modest.

As well, candidates tapped by pundits in Britain as potential successors to May are fairly known quantities and each is seen as bringing some baggage to the table. Unlike Rudd in Australia, none has led at the national level.

"I don't think there's any appetite in the country for a new general election and I don't think there's any demand amongst my colleagues for a leadership election, either," longtime MP Graham Brady told BBC News on Monday.

May has stayed on the tightrope so far by firing two key advisers and reshuffling her cabinet to keep all factions of the party on board. Michael Gove, who challenged May in a bid to succeed David Cameron as leader last year, is back in the cabinet.

With files from Reuters