Jack and Gill went up the Hill
To fetch a bottle of water;
Jack fell down and broke his crown,
And Gill came tumbling after

First appearance in a toy book, c. 1783. Baring Gould suggested that the verse preserves the Scandinavian myth of Hjuki and Bill, who were caught up by Mani, the Moon, as they were taking water from the well Byrgyr, and they can be seen to this day in the Moon carrying the bucket on a pole between them.
From Comparative Studies in Nursery Rhymes, by Linda Eckenstein, 1906.

Most of these tales behind children's verses are speculation and fantasy, appended during Victorian times.
The suggestion that Jack and Gill were Louis and Marie is more likely, but wouldn't children have played guillotine games rather than make up a simple little rhyme about the beheadings?

Now my interpretation is that Jack and Gill found the hiding place of their old man's booze, got drunk and lost their equilibrium.