This "nursery rhyme" is actually the first verse of a longer poem that was written by William Miller (1810-72). The poem was first printed in a collection published by David Robertson in 1841. Here's the whole poem:

Wee Willie Winkie rins through the town,
Up-stairs and doon-stairs, in his nicht-gown,
Tirlin' at the window, cryin' at the lock,
"Are the weans in their bed? - for it's now ten o'clock."

Hey, Willie Winkie! are ye comin' ben?
The cat's singin' gay thrums to the sleepin' hen,
The doug's speldered on the floor, and disna gie a cheep;
But here's a waukrife laddie that winna fa' asleep.

Onything but sleep, ye rogue! glow'rin' like the moon,
Rattlin' in an airn jug wi' an airn spoon,
Rumblin' tumblin' roun' about, crowin' like a cock,
Skirlin' like a kenna-what - wauknin' sleepin' folk.

Hey, Willie Winkie! the wean's in a creel!
Waumblin' aff a body's knee like a vera eel,
Ruggin' at the cat's lug, and ravellin' a' her thrums, -
Hey, Willie Winkie! - See, there he comes!

Wearie is the mither that has a storie wean,
A wee stumpie stoussie that canna rin his lane,
That has a battle aye wi' sleep before he'll close an ee;
But a kiss frae aff his rosy lips gies strength anew to me.

William Miller.

The illustration comes from Little Bo-Peep, A Nursery Rhyme Picture Book illustrated by Leslie Brooke (1862 - 1940) and published around 1922.

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