Hey, Kev. Not familiar with Kenney Ball nor Acker Bilk, but Brubeck sorta changed the rules when he changed the meter such as in Take Five.
I always called BeBop frantic music, and never cared for it, but West Coast jazz is the cool type.
As for your question about traditional jazz, I am not certain how one may classify it, but improvisation most certainly is a characteristic, and it's rather like playing around the melody. Syncopation is also a characteristic of traditional jazz.
Having been a vocalist, all I know is what I sang. Mostly up tunes and jazz ballads.
Letty your reply is more than welcome, but now I'm even more baffled than when I started, let me try and put this another way, you know when you listen to music, afterwards you can whistle the tune(for example Moon River)
Moon river, wider than a mile
I'm crossing you in style someday
dream maker you heart breaker
wherever you're going I'm going your way
Now you can hum that or sing that to yourself because it has a tune to it, what I'm asking is which jazz is it where every musician plays (improvises if you like does whatever he wants to the point where afterwards you couldn't hum it to yourself because there isn't any one rememberable tune, it's just a cacophany of sound