I’m not sure when King Arthur is supposed to have lived by the latest group of scholars coming down the pike.
If he battled the Roman invasion of 43 AD, he would have spoken a Celtic language. His language would, unfortunately, be wiped out by the later invasions of Germanic tribes after the Romans withdraw their armies.
If he was fighting invasion by Germanic tribes around 410 AD, when the Romans told the Britons to see to their own defense, he likely would have spoken some Celtic language (Welsh, Scots and Irish Gaelic, Manx, and others that have not come down to us even in writing are all examples). Or he might have spoken a Germanic dialect, a Rune says.
If he fought Vikings and Danish invasion before the treaty of 886 and the establishment of the Danelaw, he would have spoken the language of the Saxons, a Germanic dialect that had largely wiped out the Celtic languages except in some far-flung regions and in place names.
If he fought William the Conqueror in 1066 and resisted the Francophone Viking descendents from Normandy, he would have spoken Old English, the language formed from the mixture of Old Norse of the Danes and the Germanic of the Saxons. Old English was the language Beowulf was told and written down in.
After William the Conqueror made French the official language, Old English absorbed its vocabulary and morphed into Middle English. Placing King Arthur in the Middle Ages means he would have spoken French (at least in courtly matters), and his common serfs would have spoken Middle English, like Geoffrey Chaucer.
Putting him in the 1400s or 1500s places him in the middle of the Great Vowel Shift and the transition between Middle English and Early Modern English, the language of Shakespeare. Of course, at this time, there was a wide variation in regional dialect even in the span of a few miles. It would take Ben Caxton’s establishment of the printing press in London in 1474 to make the London dialect the standard across the country and edge out the competition, something that still hasn’t completely happened.
If you’ll notice, the first book published in English was made right at the height of the Great Vowel Shift. This means that the phonetic spellings that Caxton took for granted would very soon be rendered incomprehensible and downright bizarre. ‘Knight’, for example, was once pronounced as spelled. English orthography comes from the fact that spelling was frozen when pronunciation was changing radically.