"ring around the Rosie" is about the great plague. individuals with the disease would be usually red with fever. the stanza "pockets full of posies" explains the fact that people during this period when walking about would put very perfumey flowers in their pockets to hide the smell of the body outside. "Ashes, ashes, we all fall down." is about how there were so many people dying because of the plague that there were not enough graves to bury them so they cremated them. "we all fall down" is about how sometimes people who had the plague would just fall on the street and die. furthermore, this also references the point that people would just die.
hope you learned something :)