Six months ago, as Britain tried to come to terms with the result of the EU referendum, it might have been hard to believe that, by Boxing Day, Brexit would seem a trifling affair compared to the political upheaval across the Atlantic.

After all, however surprising the outcome of the referendum, there were at least logical explanations for a vote to leave the EU – in addition to the less rational and the downright nasty. When it came to Donald Trump's presidential election victory, logic appeared to be defied at every turn. Seven weeks on and Trump’s impending move to the White House still feels incomprehensible.

The early hope among those fearful that he would put campaigning slogans into practice was that a Trump presidency could not possibly be as bad as a Trump candidacy. This political newbie would surely be caught up by the machinery of government and have his eccentricity gradually chiselled away. Yet even in the past few days, the signs seem to suggest that Trump is as Trump does, and that there may be little stopping him.

Most obviously, Trump appears to be convinced that his unorthodox use of social media as communications strategy – including in respect to major matters of national policy – is here to stay. That poses a challenge for political allies and opponents alike, and has the potential to rewrite the rulebook entirely as far as global diplomacy is concerned.

Trump's tweet last week about the need for America to expand its nuclear capability was the most obvious example yet both of his grandstanding style on social media and of his capacity for radically changing key US policies in 140 characters or less. On this most explosive of subjects, it was a startling and thoroughly unexpected intervention. His qualification that such a nuclear expansion would only be necessary “until such time as the world comes to its senses regarding nukes” was hardly reassuring.

For the UK, Trump's rise to power is more likely to prove problematic than positive. The truth is, there is no evidence that America's next president has any understanding of or love for the “special relationship”. Trump's world is about transactions and about people he thinks he can do business with. When it comes to Britain, he has a pal in Nigel Farage and didn't think twice in seeking a key role for his partner in grotesquerie, never mind that by doing so he undermined the UK Government.

When it comes to trade, the US and Britain will doubtless remain key partners, but there is no obvious reason why the relationship will become more beneficial to the UK in a way that would make up for the probable negative consequences of Brexit. As for wider geopolitical questions, Theresa May will be on tenterhooks to see what Trump has in store. Does he really want to spark a nuclear arms race? Does he want to take on China? Will he and Putin continue their bromance – and, if so, what does that mean for Russia's neighbours in Eastern Europe? The UK, out of the bosom of the European Union, must work out who its friends really are.

One particularly key question for Ms May arises specifically from Trump's call for an expansion of America's nuclear arsenal: should Britain too look to rebuild its military capacity? Since the end of the Cold War, cuts to our armed forces have, rightly, been significant; the idea of the UK being involved in a conventional, large-scale conflict having seemed increasingly remote. But Trump's apparent attitude to Nato, his potentially laissez-faire approach to Russian expansionism, and his antipathy towards China have all, at a stroke, made the world more dangerous.

Numerous military figures have previously suggested that the downsizing of Britain's fighting forces has gone too far. In a memo written before he retired in April as head of Joint Forces Command, General Sir Richard Barrons cautioned the Defence Secretary, Michael Fallon, that it was only in the field of counter-terrorism that Britain's military plans were up to date. “Capability that is foundational to all major armed forces has been withered by design,“ he noted. There are more reasons to feel anxious about that warning now than there were then.