Well, whistles has been found in Neanderthal occupation layers, so whistles would be a safe bet.

Drums go way back. The drum we call a dumbek has been found in the Czech Republic c. 1600 BC and in Iraq 2600 BC or thereabouts.

I saw an ancient Greek soldier on a frieze holding a shield which bore a striking resemblance to a bodhran, so why not use a bodhran?

Harps go way, way back. I just saw a sculture from the Fertile Crescent (court of Nebuchadnezzer or some such) showing a harper at work.

Europe is only about 700 miles from one end to the other, so we can safely assume that all these had spread to Arthur's realm by the early middle ages.

You might see what instruments the great museums such as the British Library, the National Museum of Ireland and the Victoria and Albert have in their collections.(These museum names are only approximate.)

The others are right that almost nothing is known of the secular music (as opposed to the instruments) of Arthur's day. If I were you, I would listen to Celtic music for a while. Then I would play tunes entirely on the black keys of a piano for a while. Stir well, then notate the music that flows into your mind.