If the monarch will not speak when the nation's very identity is at stake, then there is no point in even having a monarch.
The Queen can only intervene in anything approaching a public way if Parliament is acting outside the terms of the British constitution. They're not, so she cannot publically intervene.
But that doesn't mean she isn't doing anything. In private, the Queen has the right to be informed about government policy, to be consulted about government policy, to encourage the government, and to warn the government. There are almost certainly matters that British governments have chosen not to present to Parliament because the Queen told them that if they were presented, she would act - and that includes matters relevant to this type of discussion. But convention means that nobody publicises this. It is private.
If a public conflict ever developed between the Queen and Parliament, Parliament would be likely to win, because that is the way the system is deliberately balanced in modern times, unless the Parliament was acting outside of the constitution. Her Majesty exerts influence behind the scenes for that reason, and while we may not see it directly, without it, Britain would be a different place than it is today. Tony Blair would have disestablished the Church of England. Britain would be on the Euro with the pound abolished. And those are just two of the most obvious differences.
The Queen has to work within the limits laid down by constitutional convention. And she does.
But there must come a time when the monarch speaks out publicly against a rot that threatens the nation. I'm not talking about taking extreme action to close Parliament or anything like that. My hope is just that the monarch would speak out publicly, in measured tones, in an attempt to change the country's direction.
Perhaps Britain is not at that critical point, yet. Or perhaps, as a Yank, I'm being too hopeful as to what any monarch would be willing to do. At any rate, thanks again for your informative reply.