My resource on Mother Goosery rhymes is THE ANNOTATED MOTHER GOOSE, by Wm. S. Baring-Gould. Wonderful book, although sometimes he’s a bit skimpy on info, but it may be that there’s not much factual info available. Anyway, he pretty much backs up PapaBear.
Baring-Gould says that Old King Cole appeared in print for the first time around around 1708. However, it is known that the rhyme (and others) were composed and sung for a long time first being printed in a nursery rhyme book. There is a 1729 History of Great Britain that mentions three kings of this name, so probably more Britons in that era were aware of the ancient kings than we’d find today. The King Cole of the jingle is believed to have reigned in 3rd Century. He was supposedly a brave and popular man who ascended the throne on the death of Asclepiod. There is a large earthwork, assumed to be a Roman ampitheatre, at Colchester, popularly known as “King Cole’s kitchen.”
His daughter was reportedly well-skilled in music (according to Geoffry of Monmouth), so he might not have always needed his fiddlers three.
“Cole” meaning cabbage, does come from the Latin colis (combined with the Dutch word sla meaning salad, to give us coleslaw). Baring-Gould does not discuss whether King Cole was therefore a satiric rhyme about some King being called a Cabbage. However, the existence of three actual kings named Cole seems to rule out your fanciful (but funny!) idea.
On the topic of Cole, Sir Henry Cole is associated with the invention of the Christmas card, in 1843. One of the joys of working for Cecil is the interesting facts one finds on the way to looking up other (unrelated) stuff.