BREXIT, it seems, will be a “great thing” for the UK – at least it will be according to President-elect Donald Trump. In an interview with the Leave campaigner Michael Gove, Mr Trump dismissed the fall in sterling and said the UK was doing fine; he also promised a quick trade deal between the US and the UK, with some suggestion from his supporters that the deal could be signed within three months of Mr Trump taking office.

In the real world, it is rather different. For a start, no formal talks can take place between the UK and the US until after Brexit. Number 10 – which this week gave its strongest signal yet that Britain is leaving the single market and customs union – also played down Mr Trump’s comments, saying Britain would respect its obligations as long as it remained a member of the EU – which means no deal within three months or anywhere like it.

The prospect of a US/UK trade deal also needs to be put into perspective. It is important that Britain strikes a strong, beneficial agreement with the United States, but if the American election taught us anything, it is that there’s a difference between what Mr Trump says and what he does.

Writing in The Herald today, Professor Phillips O’Brien of the University of St Andrews, says Mr Trump’s team will approach any deal with the UK from the point of view of US self-interest and will have an upper hand from the start because of the economic imbalance between the two countries – US/UK trade is a big part of British commerce but a small part of America’s. The bottom line is America needs the deal much less than Britain does.

The reason Britain needs the kudos of a good deal is because Theresa May’s strategy is that Britain will leave the EU and “embrace the world”. Her speech today – the first real indication of the direction of travel after months of vagueness – is about building, as she puts it, an independent, global Britain – in other words, a Britain that is unequivocally out of the EU. She does not want a partial or associate membership of the European Union or anything that leaves us half-in, half-out. She wants out. She wants a hard Brexit.

Mrs May also appears willing to accept the disastrous economic consequences of hard Brexit because the be all and end all of her strategy is control over immigration. As for the constitutional consequences, the PM is backing the Scottish Government into a position that makes it extremely hard, if not impossible, for it to achieve its aim of retaining some of its relationship with the EU. The EU expert Kirsty Hughes believes the aim of keeping Scotland in the single market cannot work in practice if the UK leaves; she also points out that were Scotland to join EFTA as the Scottish Government would like, it could not also be a member of the customs union.

On the face of it at least, the Scottish Government still believes its plan could work and has reiterated that the Prime Minister has promised the give the proposals serious consideration. But Mrs May’s speech tells a different story. The UK is heading for a hard Brexit, which Donald Trump reassures us will be great. All the signs are that it will be exactly the opposite.