EMILY (showing Lorelai a hat): Oh, isn’t this lovely?
LORELAI: Oh, yeah. As soon as we have her crowned Queen of England we’ll give it to her.
The throne of England actually came to an end in 1707; after this the kingdoms of England and Scotland merged into the Kingdom of Great Britain. In 1801 this merged with the Irish kingdom to become the Kingdom of Great Britain and Ireland. After most of Ireland left the union, its name was changed to the Kingdom of Great Britain and Northern Ireland, otherwise known as the United Kingdom. The last Queen of England was Queen Anne (1665-1714; became queen in 1702).
The Queen of the United Kingdom is Elizabeth II (born 1926), which was also the case in 2000. By tradition, the Queen (like other royals and the upper classes generally) always wears a hat in public during formal occasions. That’s probably why Lorelai connects hat-wearing with royalty. Her opinion is not shared by Amy Sherman-Palladino, who is a notorious wearer of hats.