The BBC piece is not always good history -- there are a few real howlers in the commentary -- but I suppose that's part of the charm of television.
Þis kyng lay at Camylot vpon KrystmasseWith Old and especially Middle English, you can often work out the meaning approximately by sounding out the word, remembering that "Þ" or "þ" is a "Th-" sound. The poem will sound archaic, but only a few words have passed completely out of the language. One of these is "tulkes," which is translated as "fighting man" or "soldier." Tolkien gives "tulkes" as "knights," but then translated "kniȝtes" as "lords," probably simply so as not to repeat himself. Tolkien appears to me to have adapted "tulkes" for the name of his Valar of might and prowess, Tulkas the Valiant, who laughed in war so that Melkor fled before him.
With mony luflych lorde, ledez of þe best,
Rekenly of þe Rounde Table alle þo rich breþer,
With rych reuel oryȝt and rechles merþes.
Þer tournayed tulkes by tymez ful mony,
Justed ful jolilé þise gentyle kniȝtes,
Syþen kayred to þe court caroles to make.
Read more about "Sir Gawain and the Green Knight" as a Christmas poem, if you like.