It is generally agreed by historians that Ring-A-Round-A-Rosie related to the Bubonic plague.

Ring around the rosies, 

A pocket full of posies, 

Ashes (A-tishoo)
Ashes (A-tishoo)
We all fall down

There were many plague epidemics beginning in 1348.  A very devastating epidemic

hit England in 1665. Children would join hands and sing this rhyme, possibly as a superstitious ritual to protect themselves against the scourge.

A person would break out in large round red welts  (rosies). Flowers would be

put near the person to reduce the bad odor (posies).  The person would sneeze a good deal (A-tissue--is the sound of sneezing); most people died of the plauge (we all fall down.)  It was best to burn the bodies to cut down on the contagion (Ashes).

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Who might Little Jack Horner have been? 

A legend dating back to the 19th century claims that Jack Horner was actually someonenamed Thomas Horner who lived during the reign of Henry VIII which began in 1509 and ended when Henry died in 1547.  It was during the period 1536-1541 that Henry VIII ordered the dissolution of the Catholic monasteries, making England a Protestant country.  Thomas Horner was the steward (servant) of the last Abbot of Glastonbury. The legend has it that prior to the destruction of the Glastonbury Abby, the Abbot sent Horner to London. Horner had a very big Christmas pie in which were hidden deeds to a dozen manors. It is thought that during Horner's journey he opened the pie and took out the deed of the Manor of Mells located in Somerset. This manor had a lead mine and plum is a shortened version of the Latin word Plumbum, which means lead.  The descendents of Horner have continually stated that this legend is untrue. However, the symbolisms  can be interpreted in a way that fits the legend.

Little Jack Horner, sat in the corner,
Eating a Christmas pie.
He put in his thumb, and pulled out a plum,
And said, "What a good boy am I!" 

He stopped on his journey (sat in a corner),  opened the Christmas pie ( eating a 

Christmas pie). He pulled one of the deeds out which had the lead mines ( He put in his thumb and pulled out a plum), And said,"What a good boy am I..( How smart I am to get this property). 

Old King Cole

 Legend attributes a few historical figures  as Old King Cole, but the most popular was Coel Hen, an ancient Welsh leader--originally one of the last Roman military commander. It is suggested that he turned his command into a kingdom.

Old King Cole
Was a merry old soul,
And a merry old soul was he;
He called for his pipe,
And he called for his bowl,
And he called for his fiddlers three.

Every fiddler, he had a fiddle,
And a very fine fiddle had he;
Twee tweedle dee, tweedle dee,
went the fiddlers.
Oh, there's none so rare,
As can compare
With King Cole and his fiddlers three!