"The Queen Backs Brexit"?

As a Brexiteer myself, I doubt that very, very, very much.

Her almost immediate ancestors thought that the North Sea was a lake in Germany (indeed, her father will have been taught that in his childhood), and her more distant ancestors thought that the Channel was a lake in France, the Throne that they really wanted. 

The point of the monarchy is very largely to enmesh us in various levels of political union with the Continent. 

The Queen is old enough to remember relatives born and raised in Britain who, like Queen Victoria before them, spoke only German at home, and English with heavy German accents outside. 

No monarch since James II has had an English spouse, unless you count Mrs Fiztherbert. No monarch since Queen Anne has had an English mother, unless you count the native German-speaking Queen Victoria.

James's wife, who was Anne's mother, was herself an aberration, the first English spouse of a monarch since Catherine Parr, and the first English mother of a monarch since Jane Seymour.

The future George VI married a Scotswoman because he was not expected to inherit.

The late Queen Mother and Queen Mary were both born in London. But one was universally regarded as Scots, including very fiercely by herself, and the other as German, likewise.

The next two monarchs have English wives, but that may very well turn out to be a blip.