The Noble Duke of York

I started in Wikipedia land, where I learned it was originally "The Grand Old Duke of York." It's modern form was published in 1913, but the rhyme could date back to 1642. Time for more research!

Off to the ever-reliable Oxford Dictionary of Nursery Rhymes by Opie and Opie, where a description can be found on page 442. (The latest edition of the book is now unavailable from Archive.org, but the older edition is borrow-able at this time!)

On to the next source, The Traditional Games of England, Scotland, and Ireland (Vol I of II)

       With Tunes, Singing-Rhymes and Methods of Playing etc. by Alice Bertha Gomme and published in 1894. It states the following: (Thank you Project Gutenberg!)

Find the Ring

O the grand old Duke of York

He had ten thousand men,

He marched them up the hill ago

And he marched them down again.

And when they were up they were up,

And when they were down they were down,

And when they were half-way up the hill

They were neither up nor down.

—Sheffield (S. O. Addy).

A ring of chairs is formed, and the players sit on them. A piece of string long enough to go round the inner circumference of the chairs is procured. A small ring is put upon the string, the ends of which are then tied. Then one of the players gets up from his chair and stands in the centre. The players sitting on the chairs take the string into their hands and pass the ring round from one to another, singing the lines. If the person standing in the centre can find out in whose hand

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 the ring is, he sits down, and his place is taken by the one who had the ring. The game is sometimes played round a haycock in the hayfield.

Miss Dendy sends a similar rhyme from Monton, Lancashire, where it is known simply as a marching game. For similar rhymes, see Halliwell’s Nursery Rhymes, p. 3.

See “Paddy from Home,” “Tip it.”

The next source mentioned is Golspie; contributions to its folklore by A. and B. Cumming [and others] collected and ed., with a chapter on 'The place and its peopling', by E.W.B. Nicholson, published in 1897. "Pigges Corantoe" is mentioned again, but, alas, is not able to be found.

Arthur Rackham's "Mother Goose" source, (mentioned in Opie's Dictionary) was not findable.

"100 Singing Games" by Kidson seems to also suffer the same fate.

"Less Familiar Nursery Rhymes" by Graves only allows a snippet view of the table of contents. You can see "The Duke of Cumberland" but can't view the poem.