There are lots of references in canon to historical monarchs, and the wizards and witches who were an active part of their community:
Merlin, King Arthur and the Knights of the Round Table are part of magical legend. His entry on a Famous Wizard Card confirms this:
Most famous wizard of all time. […] Part of the Court of King Arthur. (King Arthur once ruled the land that is now part of England.)
The cards also make reference to Morgana (or Morgan le Fay) as the Queen of Avalon and half-sister of Arthur.
As well as Merlin, we also learn from Pottermore that Sir Cadogan was also part of the Round Table, despite being excised from most Muggle versions of the tale.
Nearly Headless Nick was part of the royal court of Henry VII. A footnote in Tales of Beedle the Bard refers to him thusly: “Sir Nicholas de Mimsy-Porpington (a wizard at the royal court in his lifetime)”. He died in 1492, and Henry VII reigned from 1485–1509.
The Malfoys got their manor at the hand of William I. The Pottermore entry on the Malfoy family describes the first Malfoy to live in Britain as so:
Having rendered unknown, shady (and almost certainly magical) services to King William I, Malfoy was given a prime piece of land in Wiltshire, seized from local landowners, upon which his descendants have lived for ten consecutive centuries.
Their Pottermore entry also explains how they liked to rub shoulders with the Muggle aristocracy, and opposed the Statute of Secrecy because they’d lose these privileges, so we can assume William I was not the last monarch they met.
There are other minor references – Queen Maeve, who educated children in magic before Hogwarts was founded – or Hengist of Woodcroft, who allegedly founded Hogsmeade and shares a name with a Saxon king – but nothing substantially different.
So it seems that in the past, it was quite common for powerful wizards (and witches, to a lesser degree) to rub shoulders with the monarchy. This seems to have changed with the Statute of Secrecy, after which I can’t find anything describing royal-magical relations.
The third WOMBAT test proposes this theory for why the Statute of Secrecy originally came about:
Failure of Ministry of Magic Delegation to Muggle King and Queen (William and Mary) begging for protection under Muggle law
If this really was the case, then it means magical services to the Crown would have been swiftly withdrawn. They would still have known about the magical community, but be denied access to or control over it. And I can’t see the Ministry relinquishing that control.
So I think the modern Crown’s relationship with the magical community is probably similar to that of the Prime Minister: they’re kept informed of important events, but don’t have any meaningful influence.