Where does the idiom "Queen Anne is dead!" come from?

The OED attests this meaning of the phrase to 1798:

G. Colman Heir at Law i. i. 6  What will they hear but what they know? our story a secret, Lord help you!—tell ’em Queen Anne's dead, my Lady.

It cites also a usage of the phrase from 1770, but that example doesn’t seem to intend the same connotation of “old news”. Interestingly, though, it cross-references us also to Queen Elizabeth’s dead, given with the same meaning, and attested from as far back 1738 (but rarely since then):

Swift Compl. Coll. Genteel Conversat. 5 Why, Madam, Queen Elizabeth’s dead.

Looking at this in context, it seems to be roughly the same “old news” meaning, assuming the conversation in the book is supposed to be contemporary: Elizabeth I died in 1603. For comparison, Queen Anne died in 1714, so this wouldn’t have been quite so old in 1798 as Elizabeth’s decease was in 1738.

A bill to restrict freedom of worship was due to be passed, but before it could receive the royal assent the queen died. The message was therefore sounded round the nonconformists 'The Queen is dead', the result being that they were safe. It was as a result of this that Isaac Watts penned the well known hymn (a paraphrase of Psalm 90) Our God (as originally written) our help in ages past.

Queen Anne was the daughter of James II of England (a Stuart), who was deposed as king by the "Glorious Revolution" of 1688 that brought his daughter Mary, and her husband, William of Orange, to the throne as William and Mary. James had been Catholic. William and Mary were Protestant. After Mary and then William died, Mary's sister, Anne, was named Queen. She, too, was a Protestant, but it was known that James' son and Anne's brother, a/k/a the Pretender, had supporters who wished for the restoration of the Stuarts and possibly Catholicism.

In 1714, Queen Anne was known to be in ill-health. She had no heirs because her children had all died in childhood. Her official successor was to be her nephew, the son of her sister Sophia, who had married a German potentate. That nephew, George of Hanover, was German, not British. This state of affairs had the country on edge, as it was anticipated that, upon Queen Anne's death, a civil war could break out between the factions of the Stuarts and the Hanovers.

When Queen Anne finally died, her death was not announced until it was clear that George of Hanover would become George I of Great Britain and that there would be no war. By the time of the official announcement of the Queen's death to the public, everybody who mattered already knew that she had died.

And so "Tell 'em Queen Anne's dead" became the equivalent of the modern, "tell me something I don't already know."