In researching this answer, I was surprised to find that the word queendom was used in several contexts. I have traced three kinds of usage in relation to England or the United Kingdom: referential usage, usage comparing "United Queendom" to "Kingdom" explicitly, and usage punning on queen in reference to gay men.
Referential
Some uses seem merely to acknowledge that the ruler in power is a queen. In the poem "The Prophecy, 1703," (in Poems on Affairs of State, 1716) the punchline of the poem is that after a bunch of strange things happen in England and on the international stage,
England shall be, or I'm an Ass
The strangest Queendom ever was.
The Works of Mr. Edmund Hickeringill (1716) seems similarly plain:
Anno 1500, when after the Death of Queen Mary, (in this Queendom of England, her sister Elizabeth succeeding) ...
In both cases, the usage refers to a moment when a queen was currently reigning. As it happens, I could find similar usages in later reigns, though they are much less common than "Kingdom":
A national law, enforcing the allotment of one rood of land for a garden, at a fair rent, to all who should apply for it, upon all parishes throughout the United Queendom, would, in my opinion, greatly relieve the distress of the working classes. (The Fleet Papers, 19 November 1842, p. 2)
Elizabeth, as we have seen, had a stern apprenticeship to Queendom. (Sir Arthur Salusbury MacNalty, Elizabeth Tudor: The Lonely Queen, 1954)
People asking why it isn't United Queendom
The most common results in recent decades use the United Kingdom as an example of a pecularity: why is it not "United Queendom"? They may approach the question from a linguistic sense or a more explicitly feminist one. For instance:
There is still no 'United Queendom' - nor yet any sign of a 'British Republic'. All this is particularly remarkable given the recent preponderance of long-lived female monarchs (Victoria and Elizabeth II) and the recurrently shaky position of the British monarchy as a whole. (Rob Pope, Studying English, 2013).
... all I knew about the UK was that they had a queen (so why wasn't it called the United Queendom, I wondered?), and they had a lot of tanning salons. (Elizabeth Ezra, Ruby McCracken: Tragic Without Magic, 2017).
And why aren't we called 'the United Queendom' when we have a queen on the throne? What's that all about? (Onjali Rauf, in Pause for Thought, 2016).
A Little More Gay
As the word queen has been applied and in some cases reclaimed by gay men (see "queen, n.," def. 13), the term has also been used in reference to the United Kingdom in a punning sense:
In the United Kingdom, the gay traveller is seen as a lucrative market segment such that in 2006 VisitBritain, which is the official tourist agency for the United Kingdom, launched a marketing campaign that was aimed specifically at the gay travel consumer. In its Internet advertising message, VisitBritain declared: 'Welcome to the United Queendom of Great Britain ... with our proud gay history, cutting edge culture and fashion, flamboyant cities and pulsating nightlife, isn't it time you came out ... to Britain?' (Donna Chambers, in New Perspectives in Caribbean Tourism, 2008)
Shouldn't it be called / The United Queendom / Or / Does that sound / Too much / Like a gay bar? (Giles Ekins, Back to Basics and Other Stories, 2022).
