Theresa May might be riding high in the opinion polls, with five surveys giving the Conservatives a lead of between six and 16 points over Labour. But there is little sense of a honeymoon in Downing Street – not least because the imminent divorce from the European Union eclipses everything else.

The first of what will be many Cabinet skirmishes over what Brexit means has broken out. Downing Street has slapped down Liam Fox, the International Trade Secretary, for suggesting that the UK should leave the EU’s customs union, which he believes would make it easier to secure new trade deals with non-EU countries. Number 10 is adamant that no such decision has been taken – a sign that May and Philip Hammond, the Chancellor, may take a different view.

Both want “the closest possible economic ties” with the EU after the UK leaves. Hammond’s pivotal role has been underestimated in the analysis of May’s new Cabinet. Having been Foreign Secretary for two years, he is well versed in EU matters. The Treasury is a powerful player in Whitehall and will be anxious to defend the City of London’s position in the talks, amid fears that it will lose business and jobs to Paris, Frankfurt, Amsterdam and Dublin after Brexit.

UK's May pledges close ties with Germany in Merkel meeting

Attention focused on May’s surprise appointment of “The Three Brexiteers” – Boris Johnson as Foreign Secretary, David Davis as Brexit Secretary and Fox. Some Tories saw it as a brilliant move to let the Brexiteers sort out their own mess after they had 57 varieties of what Brexit would mean during the referendum campaign. Others viewed it as a clever insurance policy: if the negotiations with the EU go badly, the Brexiteers would get the blame.

Both theories are wide of the mark. May is Prime Minister and the voters will expect her to honour her pledges to respect the referendum decision and, in her words, “make a success of it”. May cannot pass the buck. Despite the headlines about Boris, the most significant reshuffle announcement was in the small print: May will chair the all-important Cabinet committee on Brexit.

The Prime Minister is certainly not enjoying a honeymoon with the hardline Brexiteers on the Tory backbenches. They are already crying foul after EU officials floated the idea of the UK winning a temporary, seven-year “emergency brake” on EU migration and keeping its single market access. About 30 Tory Europhobes could make life very difficult for May, given that her Commons majority is just 12. (Then there are the ministers she sacked and the backbenchers she overlooked to promote allies and newbies.)

Ardent Eurosceptics suspect that May and Hammond want “Brexit-lite”, and so are demanding a “hard Brexit”. They argue that the public voted to leave the single market and end EU migration, and worry about May’s talk of securing “some controls” over migration.

Their anger is premature, since May has not had time to make the crucial judgement on the trade-off between single market access and being able to limit EU migration. She said in Rome on Wednesday that she has an “open mind” about the UK’s trading relationship with the EU. One senior Whitehall source said: “We are waiting to know the PM’s bottom line on migration. Everything else will flow from that.”

The pressure on May will not only come from her own party. True, the official opposition is off the pitch, and landing punches on itself in the dressing room. Jeremy Corbyn has so few bodies willing to serve in his reserve frontbench team that he cannot appoint someone to shadow Davis; Emily Thornberry is shadow Brexit Secretary as well as shadow Foreign Secretary.

Brexit reactions – in pictures Show all 10 1 /10 Brexit reactions – in pictures Brexit reactions – in pictures Supporters of the Stronger In campaign look at their phones after hearing results in the EU referendum at London's Royal Festival Hall AP Brexit reactions – in pictures Leave supporters cheer results at a Leave.eu party after polling stations closed in the Referendum on the European Union in London Reuters Brexit reactions – in pictures Mr Cameron announces his resignation to supporters Getty Brexit reactions – in pictures Donald Tusk proposes that the 27 remaining EU member states ‘start a wider reflection on the future of our union’ Getty Brexit reactions – in pictures Ukip leader Nigel Farage greets his supporters on College Green in Westminster, after Britain voted to leave the European Union PA Brexit reactions – in pictures Supporters of the Stronger In Campaign react as referendum results are announced today Getty Brexit reactions – in pictures Boris Johnson leaves his home today to discover a crowd of waiting journalists and police officers Getty Brexit reactions – in pictures Leave EU supporters celebrate as they watch the British EU Referendum results being televised at Millbank Tower in London Rex Brexit reactions – in pictures Supporters of the Stronger In Campaign react as results of the EU referendum are announced at the Royal Festival Hall Reuters Brexit reactions – in pictures Supporters of the Stronger In campaign react after hearing results in the EU referendum at London's Royal Festival Hall PA

Thankfully, some MPs are trying to speak up for the 48 per cent who did not vote for Brexit. Owen Smith, Corbyn’s challenger in Labour’s leadership election, is promising a second referendum on the exit deal.

The Liberal Democrats have only eight MPs but are punching above their weight in the Labour leadership’s absence. Nick Clegg, back on the frontbench to speak on Brexit after a year out, knows what he is talking about as a former EU trade negotiator and European Commission official. He has assembled a team of experts and will publish a series of 12 “Brexit Challenge” papers posing the key questions the Government must answer. Clegg wants an informed public debate, and is rightly worried that the key decisions might otherwise depend on “who shouts loudest on the Tory backbenches”.

Clegg is not trying to overturn the referendum decision. But he believes Parliament must play a big role in shaping the terms on which we leave the EU, because people did not vote on them in the referendum. He suspects people will get a nasty shock when they realise the implications of Brexit. His first policy paper, on the single market, warns that the UK faces “a dramatic loss of sovereignty” – the very opposite of the “take back control” mantra of the Leave campaign. Clegg argues that what matters much more than tariffs on imports are the non-tariff barriers like the rules on consumer and environmental standards, over which the UK will have no say outside the EU.

The former Deputy Prime Minister believes May will have to give MPs the final say on the exit deal, because as Home Secretary she insisted on that over the less significant issue of EU co-operation on policing and justice. If she agrees, there could be some interesting votes because about 480 MPs backed Remain and only about 160 Leave.