* British, EU officials met in Brussels to discuss plans
* May had said talks would start June 19 but now in doubt
* Wounded by election, premier said open to changing tack
* "Softer Brexit" calls could focus on customs arrangements
* Business, EU workers voting with feet, not awaiting deal (Adds detail of Brussels meeting, more on British position)
BRUSSELS, June 12 (Reuters) - British officials held "talks about talks" with the European Union's Brexit man in Brussels on Monday but actual negotiations, scheduled to start in a week's time, may be delayed by political upheaval in London.
As Prime Minister Theresa May tried to shore up her authority after losing her Conservative government's majority in an election she had called to strengthen her hand in the EU talks, her Brexit Secretary David Davis said negotiations may not now start on June 19 as May had previously said.
That is because she is also trying to schedule the formal debate of her new government programme for next week and had on Monday yet to agree a final deal with the small Northern Irish Protestant party on whose support she will rely in parliament.
The top civil servant in Davis's Brexit ministry, Oliver Robbins, and Britain's EU envoy met chief EU Brexit negotiator Michel Barnier over lunch, EU and British officials said. They discussed how talks could proceed, a year after Britons voted to leave the bloc, but set no date for the start of talks on the substance of a withdrawal scheduled for March 2019.
EU officials said they were ready to talk now and handed Robbins texts of their position on two priority issues - rights for EU citizens in Britain and London's financial obligations on departure. An EU summit which May will attend next Thursday and Friday might mean delaying the launch until the following week.
May's spokesman said the prime minister, who in March set Britain on a two-year countdown to leaving the EU that included a clean break with the bloc's single market and customs union, was not changing her position on what she wants from Brussels.
But there were new calls in London for her to retreat from a position that some critics call a "hard Brexit" and seek a softer option.
NEW TACK ON BREXIT?
One senior pro-EU lawmaker who attended a party meeting with May - and who herself opposed Brexit last year - said the prime minister told them she wanted to build a "consensus" on how to pull out of current arrangements with Britain's main trading partner.
"She recognised the different views in the party on Brexit and she also recognised the need to not just reflect the party but indeed the parliament and the country," the lawmaker said.
Ruth Davidson, the influential leader of the Conservatives in Scotland, which voted heavily to remain in the EU last year and where May's party won rather than lost seats on Thursday, called for a deal that put "economic growth first" and said after the meeting with May that the Brexit policy could change.
Those advocating a "softer Brexit", notably in the world of business, are especially keen not to erect new barriers to trade, possibly at the cost of accepting more rules set by the EU in Brussels.
A source close to Davidson told Reuters she wanted a "shift in thinking" that would put less emphasis on cutting immigration, a key issue in the Brexit referendum a year ago.
Diplomats from some of the 27 other EU member states said they were willing to listen to new proposals from Britain. But many are also concerned that a sharp switch in direction could waste time to reach a deal on an orderly exit and could still fail to meet the approval of British voters.
Cancelling Brexit - an option not endorsed by Britain's main parties despite their opposition to it last year - or even stopping the clock to withdrawal in 22 months would require the unanimous approval of the 27. That could be hard to achieve.
Some Britons argue for staying in the EU'S single market but EU leaders insist that would require keeping Britain open to free immigration from the EU, accepting EU laws and paying into the bloc's budget. That is something many in Britain oppose fiercely.
One possibility for softening Brexit could be for Britain to remain, to some extent, within the EU's customs area. That could solve problems for Northern Ireland, where even May's pro-Brexit prospective allies want to keep the island's land border open to avoid disrupting a 20-year-old peace in the troubled province.
It could also allay fears of disruption to complex supply chains for manufacturers, but a customs deal on goods would do little to help London's big banking and insurance industries, which face exclusion from the EU-regulated market in services.
Finance industry chiefs told Reuters they were pushing ahead with plans to shift staff and operations to the continent since they could not afford to wait for details of an orderly Brexit deal and had to be prepared for talks breaking down.
In another sign of economic actors taking decisions based on a view Britain will be isolated from its neighbours, a medical charity said the number of nurses from EU countries registering to work in Britain - a key part of the healthcare workforce - had dropped 96 percent since the Brexit referendum. (Reporting by Alastair Macdonald; Editing by Tom Heneghan)