We know for a fact that Merlin (in the potterverse) was in Slytherin house and did indeed live during King Arthur's Court whilst also pertaining to the same enemy named Morgana. (source)
What other elements from the Arthurian Legend are held to have taken place in the Harry Potter universe?
Most of the information on Merlin/Arthuriana in Harry Potter comes from Famous Wizard Cards, and some info from Pottermore.
Harry Potter Lexicon is a good source of info on the cards, and the have the following (sourced from both the books and other Potter related merchandise:
Merlin
A Charms specialist sometimes known as "The Prince of Enchanters;" Merlin is unquestionably the most famous wizard of all time (PS6, fw). Merlin was part of the Court of King Arthur (King Arthur once ruled the land that is now part of England). He believed that wizards should help Muggles and therefore created the Order of Merlin to support laws protecting and benefiting Muggles (fw).Morgan le Fey, 'Morgana'
Also known as Morgana, this bird Animagus was King Arthur's half sister. She was a dark sorceress, Merlin's enemy, and she affected many events during her time. She was queen of the island of Avalon, and she had great skill as a healer. She appeared on one of the first Chocolate Frog trading cards Harry Potter ever saw
So, what we have in common (among the sparse HP details) are:
Existence of Merlin the wizard (though the dates are off - ~980 A.D. for HP, ~6th century for "real" Myrddin).
Existence of King Arthur and the Court of King Arthur
Merlin being King Arthur's educator.
Merlin being a "benevolent" wizard as far as relationships with non-magic-users
Merlin having a half-sister named Morgana, who was his opponent and evil.
Connection between Merlin, Morgan le Fey and Avalon (though typical Arthuriana doesn't make Morgan le Fey to be Avalon's queen).
Morgana being a Healer (same as in Chrétien de Troyes work)
Uniquely Potterish things:
Timeline as mentioned above. 10th century vs. 6th century
Merlin being in Hogwarts or Slytherin. Neither exists outside Potterverse, duh.
Existence of Order of Merlin.
Merlin being just one of the timline of wizards and witches. According to most of Arthuriana, Merlin was the last of magicians, due to Christianity supplanting ancestral beliefs of Britons and magic disappearing from the world.
Morgan le Fey being a queen of Avalon is a wash. In Chrétien de Troyes's work, Morgan is the lover of Guingemart, lord of Avalon. In other sources, she is associated with Avalon but not a ruler.
Morgan le Fey being a bird-turning "Animagus".
Adding onto what DVK has written, J.K. Rowling wrote in the Slytherin welcome letter that Merlin was Sorted to Slytherin as a first-year at Hogwarts:
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As well, Sir Cadogan, the short, pugnacious knight who guards Gryffindor Tower in the absence of the Fat Lady, originates in Arthurian legend.
‘Aha!’ he yelled, seeing Harry, Ron and Hermione. ‘What villains are these that trespass upon my private lands? Come to scorn at my fall, perchance? Draw, you knaves, you dogs!’
They watched in astonishment as the little knight tugged his sword out of its scabbard and began brandishing it violently, hopping up and down in rage. But the sword was too long for him; a particularly wild swing made him overbalance, and he landed face down in the grass.
‘Are you all right?’ said Harry, moving closer to the picture.
‘Get back, you scurvy braggart! Back, you rogue!’
*Prisoner of Azkaban - page 77 - Bloomsbury - chapter six, Talons and Tea Leaves - Sir Cadogan and Harry Potter
Sir Cadogan is said by J.K. Rowling on Pottermore to have been one of the Knights of the Round Table, although his existence as such has long been forgotten, or edited out, of Arthurian legend by Muggles over the years.
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‘Farewell, my comrades-in-arms! If ever you have need of noble heart and steely sinew, call upon Sir Cadogan!’ - Prisoner of Azkaban
According to Pottermore, Sir Cadogan was a Gryffindor at Hogwarts. He was left by at least three wives and has seventeen known children. Little chap's got to be brave to have seventeen kids!
