All of the Commonwealth Realms are constitutional monarchies, and democracies. There’s a short list of things the Queen can just wake up and decide to do. One of them is sell Sandringham House, a little palace out in the sticks that actually belongs to her because her dad gave it to her.  She could buy another one with the cash. She could go on the shopping channel and get retro popcorn makers for everyone she knows next Christmas.  She can’t sell Buckingham Palace or Windsor Castle though, because they’re not hers, they’re the monarch’s. She happens to be the monarch, but eventually that will change, and she can’t get rid of the “crown estate” because it isn’t really hers to do. The other thing she can do is appoint people to the Order of Merit.  In pretty much every other way, she is bound by the law, or by constitutional etiquette centuries old. The woman can’t even decide who shows up at most of her tea parties.
So in this situation, Boris decides to play games with Parliament and use procedural tricks to stop them doing anything about his mad ridiculous plans. He’s doing a great job of sucking Nigel Farage’s cock at the moment, so good that Nigel’s almost thinking of returning the favour. It’s very fucking obvious that no one ever voted to just randomly leave Europe, no one accepted that during the entire negotiating period, no one could stomach actually voting for the deal they got, because then it means they put their stamp of approval on this shit show.  The only thing left is a) admit it was a giant fucking disaster and could we please stay, or b) play stupid brinksmanship games that will just make it worse for everyone but that’s still a few weeks away and that means a few more weeks without being blamed.
So he trots off to ask Her Maj to push the Procedural Reset Button on Parliament, purely to waste time and avoid getting slammed in debate every day. 
What should the Queen do?  Should she think twice and defy him?
No! Because as that Scottish judge pointed out, Parliament still has the tools to deal with this in its own way, and therefore this needn’t bother the Queen.  There are two options, and time enough for both.  Parliament could kick Boris out and pick someone else to be Prime Minister.  He could go kicking and screaming and try to call for elections or tell the Queen he won’t go.  And she literally gets all her news from him, at least officially.  She knows what’s going on, but if he shows up at the Palace and says “Your Majesty, Parliament just voted to double-plus-extra Brexit!” she will say “Excellent, Boris! Where do I sign?”
And legally, anything she signs is perfect. She is Queen after all.  On the other hand, anything **he** says is very much subject to being destroyed by a judge on account of being patently illegal, or manifestly insane, or when it comes to the constitutionality of it, “That simply isn’t done!”
So we’re in a situation where the Queen is just going to roll and do her thing, which means whatever Boris tells her, but every word he tells her is very much subject to being declared illegal, or out of bounds, or “just not how it’s done.”
Lots of chances for this to still wind up in court but two things will happen: it will all be about what Boris says, not what the Queen does. And the Queen will do what Boris says.
Even to the point of losing a major court case, Boris will be expected to show up at the Palace and be the one to tell her, officially, “This is terribly awkward, ma’am, but apparently i was mistaken in the advice I gave. Would you mind terribly un-proroguing Parliament / un-calling the election / undoing whatever daft illegal thing?”  If he refused, instead of backing down / doing the honourable thing and resigning / etc., it would be an act of insanity. And it would probably quickly turn into a new constitutional precedent.  Parliament would petition Liz directly.  The custom of Tea Time with the Prime Minister would take a distant second place to “An Urgent Petition from the Members of the House of Commons and the Lords of the Realm Assembled, Entreating Her Majesty to Act Swiftly on a Matter of the most Grave Importance regarding Seditious Acts by the Prime Minister.”  
And then for the first time in a couple of centuries, the Queen would Rule, and not just Reign.
So yeah she is still there as a democratic and legal safety valve if the Prime Minister goes off his rocker.  But honestly there’s still a long way before that would happen.Parliament still has time to dump Boris, overrule Boris (already done as of this writing), sue Boris in court (still proceeding; very likely more to come) and frankly they could fit in an election, someone could campaign on stopping Brexit or delaying Brexit, and then that’s what would happen.  
There’s still lots of time before we need to reach for the Commonwealth Panic Button and involve the Queen directly. There’s still lots of time for Boris to play these games, and the reason there’s so much sangfroid and cooler heads prevailing, is because the judges know it, and so does the Queen.
If they do anything different at all, they’re maybe giving Charlie an extra briefing on current developments in case the Queen takes ill and he has to step in as Counsellor of State. They do need everyone on their toes at the moment, but it’s still business as usual.