I have no clue how modern monarchy works, but can she abdicate, officially distance herself from her royal family somehow so that she becomes a normal citizen? Maybe become a
nun?
I think if a "totally random old woman we found on the street somewhere with the name of Elisabeth (what a coincidence)" stands in front of the press and delivers a state of the union speech, then the Brits will listen to her.
That way, monarchy itself could persevere, and only she would have to throw herself under the bus, figuratively speaking. (With the understanding that the UK would have to spot her a nice little apartment and some pocket money, obviously. Would be a small price to pay.)
So, I mean, let's break this down a little.
Theoretically, yes, Elizabeth II Stuart is by the Grace of God Queen of the United Kingdom of Great Britain and Northern Ireland. She can do whatever She feels like doing, She's a divine monarch!
In practice the queen, like everybody else in British politics, is heavily bound by constitutional norms, and for centuries the constitutional norm has been that the royal family really doesn't do anything, because no matter who gets elected to government she has to go along with them.
Could she violate constitutional norms? Sure. It might even work -- see Emperor Hirohito during the Japanese surrender in World War II. But she'd have to be very confident of the rightness of her action and the safety of her ground, or there would be intense backlash -- not in the form of protests or lawmaking but just in people refusing to work with her. Remember that it's less than a hundred years since the British government forced the king to abdicate because they didn't like his marriage choice, which is a somewhat less serious faux pas than speaking out against the result of a democratic referendum called by your government.
But if you want an example of what the consequences would be, you don't even have to look that far back. Jeremy Corbyn is violating the constitutional norms and expectations of the Labour Party leader right now, and that's why he can't muster up enough MPs to fill a bench. That's what the UK government might look like if the queen got involved in politics, until she agreed to abdicate. It's pretty unlikely that that would have any particularly good effects on the current situation. (And no, abdicating beforehand wouldn't really change the ground here. She wouldn't get punished for her actions, but that wouldn't make them useful actions.)
So that's why it's very unlikely that the queen will take any direct action. That doesn't mean she won't call David Cameron on the phone periodically to tell him that he's an asshole, but British constitutional norms are very strict about monarch involvement in politics. The main responsibility of the Queen is actually to help resolve constitutional crises caused by the violation of norms, so she really has to avoid starting her own.*
* Fun note -- in writing this post I went to look up British constitutional crises on Wikipedia to see if I could come up with some examples from after 1911. Here's the latest entry on that page:
wikipedia said:
The 2016 EU independence referendum with Scotland and Northern Ireland voted to remain in the EU but England and Wales did not. Boris Johnson had led the 'Vote Leave' campaign to victory, leading to the resignation of Prime Minister David Cameron and a Tory leadership contest. Meanwhile, Labour Party is facing a meltdown after a failed coup to remove Jeremy Corbyn by the Labour MPs, which has led to the resignation of many shadow ministers and many roles in the shadow cabinet remains unfilled. On 30th June, Boris Johnson withdrew his Tory leadership bid upon learning that Michael Gove, who he ran the Vote Leave campaign with decided to put his name forward to be the next Prime Minister. UK as a result is facing the possibility of Scotland breaking away from the union, and a power vacuum in the 2 main political parties.
Wikipedia is on the case!