The optimistic version – pushed heavily by the hard-right Eurosceptics and their many media allies – is that Donald Trump’s promise to emulate Brexit (“Plus. Plus. Plus”) and the encouraging noises from his camp about a UK-US trade deal are all good omens for warmer, more advantageous relations. It is true that the Cameron-Obama partnership was not a marriage made in heaven and, simply on the grounds of their shared gender, it is hard to believe Hillary Clinton would have reversed President Obama’s policy and put Theresa May at the front of the queue for trade and much else.

Personal relations between Ms May and Mr Trump may not, in the event, prove so very warm after all – there is little to suggest a rerun of the Gone With the Wind near-romance between Margaret Thatcher and Ronald Reagan. They are not ideological soulmates, and have little in common, it would seem.

Given all that, the suggestion – apparently rubbished on all sides – that Nigel Farage could act as some sort of “go-between” for the White House and No 10 is neither as bizarre nor as unwelcome as it first appears. If Britain wants to get the best from its relationship with the largest economy on earth, the world’s pre-eminent military and financial power and the historic arsenal of democracy, then the nation should make the most of whatever assets it has. If those include the eccentric figure of Mr Farage, and his rapport with The Donald, then so be it.

Post-Brexit, the UK needs all the friends it can get. The British, soon to be separated from their largest export market, can ill-afford to be fastidious.

Britain’s chances of a trade deal with the US – a version of the proposed TTIP (Transatlantic Trade and Investment Partnership) with the EU – may be better than they might have been under the Democrats, but that does not mean the deal would be of great value. Mr Trump stood on clear policy of bringing jobs back to the US through easing back, if not reversing, globalisation. International trade, though, is all about globalisation, and there is nothing that “special” about British exports compared with those of, say, Germany or Japan that makes them in some way harmless to the American competition. Apart from his own prejudices, Mr Trump will be crowded by all those special interests he talks about, arguing that a trade deal with the UK isn’t worth the jobs and votes it might cost in key states or strategic industries.

Besides, trade in services, or at least banking and financial services, is often a matter for individual state regulation rather than the federal authorities. It may not be possible, even with an indulgent Trump White House, to make much progress on services, on the recognition of professional qualifications and, most sensitively of all, immigration. It seems certain that the UK will not be granted the same sort of unrestricted “single market” access to the US it currently enjoys in the EU. The British should not get their hopes up too high.

Elsewhere, the whole world will be lucky if Mr Trump doesn’t do serious damage to the world economy. The dollar is America’s currency, but it belongs to the world. Sometimes literally, given the vast holdings in China, Japan, the Gulf States and other places.

The dollar is edging into jeopardy. Mr Trump’s plans for tax cuts and infrastructure investment may have many arguments in their favour; but they would, at least in the short run, widen still further the federal deficit and add to the debt mountain – almost $20 trillion (£15.8 trillion). If the world’s investors, especially in China, lose faith on the dollar and America’s capacity to honour its obligations, then the consequences would be grave indeed, for virtually everyone in the world, via their banks, insurers and pension funds, owns dollar assets. So more important to Britain, because of its crucial global implications, than any Anglo-America trade deal is to see a more harmonious and sustainable trade and investment relationship between the US and China in particular. Ms May, Mr Farage and the Brexiteers will be mere anxious spectators at that event.

We should also be wary about what a Trump administration will mean for Nato, European security and the threat of Islamist terror. Mr Trump has dropped ominous hints about the dispensability of Nato, and thus the security of much of northern, eastern and central Europe from Russian pressure and, even, territorial ambitions. We do not want to see the Iron Curtain back. If the price Europe pays for America to be friends with Russia and to defeat Isis in Syria is a more aggressive President Putin, then that is obviously unacceptable – and bad for America too.

America is right to ask Europeans to pay more towards their own defence; but wrong to assume that what happens on this side of the Atlantic does not affect America’s own security.

We can only hope, too, that a President Trump will learn to use language about Muslims that is more temperate and, frankly, accurate. Such terminology matters and one of the worst episodes in a uniformly embarrassing and dispiriting Trump campaign came when “Donald J Trump” called for a ban on all Muslims entering America (presumably including the King of Saudi Arabia, the Mayor of London and the President of Turkey). So-called Islamic State, Al-Qaeda and their allies have long hoped for and fomented a global religious war, and Western leaders have sometimes fallen into the trap of responding in kind to their terror. If President Trump does too, and is not smart enough to see that massive brutal indiscriminate retaliation for acts of terror is what Isis and the others want, then the prospects for harmony and security in the UK are also diminished.