The Queen is a sham head of state. She cannot act as a constitutional president and force rival politicians to look for ways out of a national emergency. She cannot insist that the prime minister obeys the rules, because there are too few rules in Britain and too many woozy, unenforceable conventions.

Elizabeth II’s strength came from being “above politics”. Now even that pose – how can a head of state be above politics? – has been exposed, as Buckingham Palace’s “displeasure and annoyance” at David Cameron reveals. It is not angry because he “dragged the Queen into politics” but because he revealed that she happily got stuck in without the need for dragging.

She did not slap down Cameron in 2014. She did not cry: “Never in my reign has one of my prime ministers dared to suggest that I intervene in a political debate.” Cameron asked her to tell the Scots that she did not want them to vote for independence. She was to keep it classy – she was the Queen, after all. He was “not asking for anything that would be in any way improper or unconstitutional, just a raising of the eyebrow, even, you know, a quarter of an inch”. She duly urged Scottish voters to “think very carefully about the future”. The popularity of the Queen rests on the notion that she is not political. Naturally, that made her political intervention all the more effective.

Johnson exploited the convention that the Queen does what the PM tells her, and got his way

Cameron has revealed that the tearing up of the conventions that once governed Britain did not begin with Brexit and no amount of smoke from the “constitutional experts” the British have in place of a written constitution can hide the fact. Peter Hennessy, who I like enough to suspect knows better, spoke for virtually the entire national media when he blamed Cameron and excused the sovereign. “These things have to be kept private,” he thundered, while ignoring that there was nothing private about her intervention in the Scottish referendum.

No one asked why the monarch cannot intervene in politics, as Prince Charles has done for years. At the level of prime ministers, what would stop a Corbyn administration, say, telling the Queen to use her political clout – nothing improper or unconstitutional, just a raising of the eyebrow – to help it? If the Queen were to refuse, as she is likely to, because the head of a hereditary aristocracy is no more likely to support a radical left programme than Scottish independence, could Corbyn denounce her political bias?

From the point of view of the monarch, the question is anything but hypothetical. Boris Johnson and Dominic Cummings understand that the old conventions of public life are there to be subverted. Blow away the dust and they crumble. It was convention that prime ministers resigned when they lost the confidence of the Commons. Johnson did not resign. He behaved as if he commanded a Commons majority and it had mandated him to shutter parliament for five weeks. Critics condemn Johnson’s lies to the public and, presumably, to the Queen. I accept that the brass-necked absurdity of pretending he is not trying to avoid being held to account on Brexit is his greatest insult to the national intelligence to date.

If she agrees to suspend parliament twice it will be political. If she refuses it will be political

More important constitutionally are his double standards. He is behaving as if he is a strong prime minister when he is an enfeebled politician without a mandate or a majority.

When a prime minister with a majority asks the Queen for a prorogation, it’s a simple administrative matter. Parliament approves, so no one can object. Johnson has no majority. He is using prorogation to escape the democratic control of the majority MPs. He and Cummings are the most dangerous type of men I know: wimps who pose as tough guys, and lash out at everyone who exposes them.

Anne Twomey, the leading authority on monarchical power in Commonwealth democracies, thinks the Queen had the right to say Johnson’s request to close parliament was “unconstitutional” because it was “done for the purpose of avoiding a vote of no confidence or other action by parliament against the government’s will”. In other words, he was seeking to silence MPs and deny the fundamental duty of the legislature to scrutinise the executive.

Johnson exploited the convention that the Queen does what the PM tells her and got his way. But another convention is that she stays out of politics and she has broken that. If she can break one convention, she can break another.

I am not raising a hypothetical question. The issue could explode this week. When the supreme court judges asked what the government would do if it found suspending parliament was unlawful, Johnson’s lawyers said: “Depending on the court’s reasoning, it would still either be open or not open to the prime minister to consider a further prorogation.” Cummings has said that he would happily send Jacob Rees-Mogg off to tell the Queen to close parliament a second time.

If she agrees to suspend parliament twice it will be political. If she refuses it will be political. There’s no escaping the politics. But because Britain does not have a written constitution, however, there is no politically legitimate way for her to react unless Her Majesty’s judges protect Her Majesty by tying Johnson down.

Brexit is showing that the monarchy is both too strong and too weak. It has formidable formal powers but convention dictates that it can be the puppet of a prime minister who lacks legitimacy.

Today’s impasse was bound to come one day. The trouble with conventions is that they can be twisted by any leader unscrupulous enough to insist that others must obey the conventions they break. Once the Brexit crisis is over, there will be an enormous job of work blocking up the loopholes that have allowed Johnson to abuse his power. Assuming, that is, we ever get out of the Brexit crisis and have the energy left to tackle any job of work again.

• Nick Cohen is an Observer columnist