* British finance minister sets out what divorce he wants
* EU negotiators: talks could be delayed
* Divisions deepen over Northern Irish border issue
* EU official: Not sure Britain know what it wants (Adds newspaper report of joint statement by finance minister Hammond and foreign minister Johnson, paragraphs 18-20)
LONDON, July 28 (Reuters) - Britain's progress towards life outside the European Union became more entangled on Friday, with divisions deepening over Northern Ireland's border and even the type of divorce Britain actually wants.
The uncertainty coincided with the EU's top negotiator warning that formal talks are set to be delayed, eating up more of the two-year divorce timetable.
Negotiations on the future relationship between Britain and the EU are now less likely to start in October because of a lack of progress at the initial stage of talks about the breakup, Brexit negotiator Michel Barnier has told EU ambassadors.
Britain responded that it was confident that enough progress could be made to start the second stage of talks but as Prime Minister Theresa May holidayed in Italy, her ministers engaged in a public debate about how Brexit should look.
Finance minister Philip Hammond, who opposed leaving the EU in last year's referendum and has one eye on the business community, said there should be no immediate change to immigration or trading rules when Britain leaves.
A shift to new arrangements could last until mid-2022, he said, adding he wanted to avoid a cliff-edge. He stressed that British hospitals and care homes relied as much on EU migrant workers as many businesses.
"We've been clear that it will be some time before we are able to introduce full migration controls between the UK and the European Union," he told BBC radio.
May's loss of her majority in the British parliament with a botched gamble on a snap election has prompted an apparent softening of rhetoric on Brexit. But some EU member state diplomats say it now hard to discern what Britain wants.
Britain has less than two years to negotiate the terms of the divorce and the outlines of the future relationship before it is due to leave in late March 2019. Both sides need an agreement to keep trade flowing between the world's biggest trading bloc and the fifth largest global economy.
"In the immediate aftermath of leaving the European Union goods will continue to flow across the border between the UK and EU in much the same way as they do now," Hammond said.
Britain's economy weathered the immediate shock of last year's vote to leave the EU much better than the government and most analysts had predicted.
But growth in the first half of this year has been the weakest since 2012, and earlier on Friday a closely watched consumer survey showed sentiment was its weakest in a year. Households viewed the economy as the worst in four years.
May expects what she calls an implementation phase but she has given few details of how it would look. Any such deal will also be subject to discussion with the other 27 EU members.
Hammond's tone, meanwhile, is sharply different that of some other senior ministers in May's cabinet who want a cleaner break with the EU including swift controls on immigration.
"A transitional deal will delay all the benefits of being able to control our laws, trade and borders. We need to get on with it," said Richard Tice, a Brexiteer who helped fund one of the Leave campaigns in the EU referendum.
The anti-EU United Kingdom Independence Party said Hammond's words indicated uncontrolled EU immigration would continue for years after the 2019 leave date.
Late on Friday, the Daily Telegraph reported that Hammond and foreign minister Boris Johnson, who favours taking a tough stance with the EU on Brexit, had issued a joint statement saying they were "working together to take the UK out of the EU."
It made no mention of transitional arrangements, the newspaper said.
Government representatives were unable to immediately confirm the joint statement.
It was unclear whether Hammond's proposals would become government policy though the implications could be far reaching.
The proposals could be read to mean that Britain would continue to pay into EU coffers during the transition, continue to accept EU laws and even effectively accept its four freedoms allowing free movement of people, capital, goods and services.
"I am not sure if they fully know what they want themselves," one EU official who spoke on condition of anonymity said.
The European Commission said that discussions about a potential Brexit transition period could only begin one divorce issues are settled.
Before talks on a transition can begin, the EU wants to settle three main points: the future rights of expatriate citizens, the exit bill Britain has to pay and how to avoid the reimposition of border controls between the Republic of Ireland and the British province of Northern Ireland.
Neither side has proposed a solution to the Northern Ireland issue, which remains sensitive almost two decades after a peace deal ended years of violence in the province.
Ireland is against the imposition of an "economic border" with Northern Ireland and the Irish government is not going to help Britain design one, Prime Minister Leo Varadkar said.
He was speaking after Northern Irish protestant politicians propping up May's minority government reacted with fury to a report that Dublin wants customs checks on boats and planes between Britain and Ireland rather than along its land border with Northern Ireland.
Ireland's foreign minister said no such proposal existed.
"As far as this government is concerned there shouldn't be an economic border. We don't want one," Varadkar told reporters at a briefing in Dublin.
(Additional reporting by Kate Holton, William James and William Schomberg in London, Padraic Halpin in Dublin and Philip Blenkinsop and Jan Strupczewski and Brussels; Editing by Jeremy Gaunt and Grant McCool)