Here are three examples from the second source cited. I could not locate an "on-line-version" of the Oxford text you note. For curiosity's sake....what does Oxoford say about these three?

(all are direct quotes from Nursery Rhyme History

Old King Cole

There was a British King by the name of Coel--his capital was Colchester. He married off Helena, his daughter, to a Roman general, and she became mother to the Emperor Constantine. Helena was declared a saint for her work in the spreading of Christianity. Coel's merry soul may relate to the peace in his Kingdom that was brought about by Helena's marriage.
This origin was contributed by Tracy Lightfoot. On a page called, Rhymes and Nonesense it is said that this rhyme first appeared in print in 1703. It also states that it refers to a 3rd century king.

Humpty Dumpty

During the English Civil War (1642-49) "Humpty Dumpty" was the name for a powerful cannon. It was mounted atop the St. Marys Wall Church in Colchester to defend the city against seige in the summer of 1648. (Colchester was a Parlimentarian stonghold but had been captured by Royalists and they held it for 11 weeks.) The church tower was hit by the enemy and the top was blown off. "Humpty Dumpty" fell off and tumbled to the ground. The King's men tried to fix him but to no avail. This origin is submitted by Tracy Lightfoot. She recieved her information from the East Anglia Tourist Board in England. As for the accuracy of this origin Tracy says it is still unsure. But, she has told me that St. Marys at the Wall Church was a stronghold during that seige. Also, there is a similar story in Glouchester, England. On the page, Accidents May Happen reference is made to this rhyme poking fun of a nobleman who had fallen from favor with the King. This king may have been Richard III (1452-1485) of England. From Scott comes the idea that this might refer to King Richard III. He states that: " Humpty Dumpty referred to King Richard III, the hunchbacked monarch. At the Battle of Bosworth Feild, he fell from his steed, a horse he had named 'Wall' (as dramatically rendered in Shakespeare's play 'Richard III:' 'A horse! A horse! My kingdom for a horse!'). Richard was surrounded by enemy troops in the battle, and was butchered right there, his body being hacked to pieces. Hence the final part of the rhyme: 'All the King's horses and all the king's men couldn't put Humpty together again.'" This mention of Richard III is not a new idea. And the dictionary defines "Humpty Dumpty" as being a person who is short, stout, kind of dumpy looking. If Richard had a hunchback, then he may have been called this.
According to "Encyclopedia Britannica" (1996) "As to the allegation...that Richard was a hunhback, neither portraits nor contemporary reports reveal such a deformity, although there is some indication that one of his shoulders was higher than the other. From Syd Dickenson we get a very different idea. He says he thinks that "Humpty Dumpty" refers to the tale of Charles I (Humpty Dumpty) of England. He was toppled by the Puritan majority in Parliment (the great fall). The King's army (Cavaliers) could not restore his power. Charles I was executed by the Roundheads* ("couldn't put back together again"). Since the World Book Dictionary states that "Humpty Dumpty" refers to a short, stout person this idea is very plausible.

*Roundheads as defined by the Dictionary of Phrase and Fable: "Adherents of Charles I. Those opposing Parliament party were called 'Roundheads.'"
This rhyme appears in Lewis Carroll's book "Through the Looking Glass." There is an allusion to this being a riddle, as well. RIDDLE IDEAS:

Kilkenny Cats

There was once two cats of Kilkenny
Each thought that there were two cats too many
So they fought and the fit,
And they scratched and they bit,
Till excepting their nails and the tips of their tails
Instead of two cats there weren't any.

According to the Dictionary of Phrase and Fable this rhyme came about because of a lie. In Ireland Kilkenny (a county and town) was invaded by Hessian soldiers. For fun these soldiers would tie together the tails of two cats then throw them across a clothes-line and watch them fight. One of the troopers, not liking this sport, snuck over and cut the tails off of the two cats. The cats fled the scene tailless. When an officer asked about the two bleeding tails still over the clothes-line the trooper told him that the cats devoured each other except for the tails.
From the "World Book Dictionary" I also learned that "Kilkenny Cats" is sometimes a phrase used to describe people who fight relentlessly.

This comes from Sadao Yasui. He states that the answer is an egg (Humpty Dumpty is always depicted as an egg in drawings). The basic question of the rhyme would be: "What is it that the King's men cannot put back together?"