So, with Britain completely narfing things up over the EU referendum, what can be done about it? I'm Australian so this doesn't really affect me, but I do teach politics and law so I find it interesting to think about.
Well as far as the government is concerned, not much can be done. The people have spoken and now its up the their representatives to enact the referendum through assorted Acts of Parliament.
But what about the Queen...?
The Queen holds certain powers and while most know that Her Majesty is able to sign Bills into Law, she is also able to block them... in certain circumstances.
The Queen must give her "Assent" or "Consent" to any Bill or debate in Parliament. Consent is used when an issue that directly affects the Monarch's powers or their ability to govern. If the Queen withdraws Consent then the debate ends there an then and nothing about that issue can proceed in Parliament. For example if the debate was about affecting the payment of taxes in the Duchy of Cornwall - which is a personal holding of the Queen and forms part of her regular income, - then the Queen could block that debate. The last time Consent was withdrawn was a Bill concerning the bombing of Iraq in 1999 which was introduced as a Private Member's Bill. However the wording of the Bill also removed the Monarch's authority (The Queen is Commander in Chief and the one who declares war when the time comes) over the use of military forces in any capacity, so that was inappropriate. (Some anti-monarchy MP was seemingly trying to use a short-term issue to further weaken the Crown in the long term.)
So Consent here isn't useful for the issue of Britain leaving the EU.
Assent is different. This means that the Queen refuses to sign a Bill into Law as an Act of Parliament. It's the equivalent of a veto. Now this could mean that the Monarchy blocks any of the Acts of Parliament that will be necessary to pass before Britain can leave the EU. Britain will remain in the EU as a result.
But will the Queen do it...? Probably not.
It would spark a major constitutional crisis and probably lead to the abdication of the Monarch. Not something the Queen has any intention of doing as Britain has enough problems right now without that happening as well, even though she would be perfectly within her right to withdraw Assent.
So Britain, you've made your bed and now you're just going to have to lie in it.