Because of a lack of sources to address your question, the most accurate, defensible answer is that no one can really know much about Arthurian folk traditions through most of the medieval period. Arthurian tradition was at once both a folk tradition and a literary cycle (with what was likely a great deal of influence from one to the other and back again!).
When Geoffrey of Monmouth (c. 1095 – c. 1155) wrote about Arthur in one of the more extensive early treatments, it seems that he was likely drawing on folk tradition, but we cannot be certain of that. There is a dispute as to whether Geoffrey was Welsh or Cornish, but in his lifetime, there wasn't necessarily much separating the two when it came to much of their traditions, so the dispute probably doesn't matter much in the context of your question. It is easiest to think about Arthurian folk traditions among the Welsh and Cornish by the time of Geoffrey, but before that, we have next to no evidence about folk belief. It is likely that it was percolating in some way in those areas before Geoffrey wrote, but there is no way of knowing anything about the specifics of this process.
Arthurian traditions were apparently the property of the Brittonic speakers of Cornwall and Wales, and it is unlikely that it diffused as a folk tradition to the English. The path of Arthurian tradition into the culture of the speakers of English was likely literary - primarily if not exclusively - at least for a very long time.
It may be possible to imagine an English Arthurian folk tradition by the late middle ages, but there is little evidence to affirm the existence of such a thing. Pre-industrial collectors of the nineteenth century were able to document English folk references to Arthur, but the more robust collections about him were still among the Cornish and Welsh: this suggests that while the English-speaking folk knew of Arthur and cast his court in the role of a setting for their stories, there was little in their tradition that were actually stories about Arthur: belief and story was most at home in Cornwall and Wales during this later period, and we can imagine that this was the case going back to earlier centuries, but we imagine those things with little evidence.
Arthurian legends were extremely popular in the medieval period. Early fantasy writer Geoffrey of Monmouth had a 'best seller' on his hands and it, in particular, caught the attention of the Plantagenets of England. During the reign of Henry II it was quite a common debate just which culture Arthur came from and who he'd been so effective at killing. Everyone with noble blood wanted to be able to add King Arthur to their own bloodline. There are even claims that Henry II led the investigation for Arthur's remains.
The story of King Arthur was in fact so popular still over 400 years later that Henry Tudor (Henry VII) named his first born son 'Arthur' with the specific intention of stirring up grand ambition and yearning for the young prince to reign over a golden age for England - that the promised king had returned (of course that didn't work out too well and we got Henry VIII instead). Henry VII's right to rule was very poor indeed so if he chose to inspire tales of King Arthur you can bet that was the strongest PR strategy he could think of.
And, of course, Edward III based his Knights of the Garter on the Knights of the Round Table concept.
I'm not well versed in any line but the Plantagenets so I couldn't comment on what the royalty of continental Europe thought but Dan Jones refers to some of this in his work on the War of the Roses and his book on the Plantagenets. Certainly Arthur was just as popular with the commons of England during this period.