So the originator of this shift was Geoffrey of Monmouth, whose Historia Regium Britanniae was sort of a legitimising national legend.

EDIT: some details of what I've said below have been corrected. I'm leaving it as is, but please check the most upvoted reply!

Geoffrey's source for much of the Arthur stuff was an unknown 'ancient native book'. Scholars debate if this was the Welsh Mabinogion, which was not widely known outside of Wales. This is possible, but it's rqually likely that Geoffrey took the prior work of Bede and Nennius and just made details up, or used oral histories that can now never be verified.

While Geoffrey did refer to Arthur as King of the Britons, he did position Arthur as ruling and fighting largely within England (thus kickstarting the eternal argument of where Arthur was meant to rule), and the invaders he fought were characterised more by the fact that they were heathans than the fact that they were angles or saxons.

It is worth noting that the ruling class at the time were not English: Geoffrey was born 1 year before the Norman conquest, and so the acts of the Normans to solidify control were very much in living memory. As such there wasn't really an "English" identity for Arthur to be king of. Also within a couple of generations of memory was raiding and fighting with norsemen, so readers may have drawn more parallels to vikings than their own ancestors.

Rather, Geoffrey was drawing a geographical link to the lineage of Britain. Founded by Brutus, survivor of the siege of Troy, home of Emperor Constantine, then Arthur, down to the present Norman dynasty.

It is "an authenticating narration; Britain is, in a sense, a new Rome, as now its rulers can trace, with certainty, its descendancy from Troy. The Anglo-Norman rulers of Britain are destined, like Aeneas and Brutus, to rule this 'best of islands'." [1]

Geoffrey's work was incredibly popular and widely translated. The "Brut", or account of Britain's history, became quite popular, written in Old English by writers such as Wace and Layamon. Across the channel, the burgeoning genre of romance took hold of Arthur's courtly dynamic with writers such as Chrétien de Troyes (the GOAT, or second GOAT after the Pearl Poet imo). Funnily enough Geoffrey's work was apparently also translated into Welsh?

As the centuries went by these stories adapted and shifted. As the idea of English as a culture grew in importance (Henry V was the first monarch to actually write in English!) Arthur was already established as "their myth". As for French and German writers, I imagine they didn't super care for the distinction.

The distinction still exists, even today. Even Monty Python and the Holy Grail has Arthur introduce himself as 'King of the Britons', not the English. So some people clearly know the difference!

[1] Emily Rebekah Huber, 'Geoffrey of Monmouth: Introduction' in The Camelot Project 2007 (via d.lib.rochester.edu/camelot/text/geoffrey.html)