tombstone,
I have looked up "The Brook" and sung it to "Yankee Doodle"! It does work, especially if you don't try to jam a verse of the poem to the song's chorus.
I mean you have to cram syllables together.
The syllables all fit almost perfectly to the melody and rhythm of the song's basic verse, not so perfectly to the chorus--although it's doable.
In terms of the poem's overall structure, there are 13 verses, and every 3rd or 4th verse, there is that repeated theme:
[Something something la la la]
To join the brimming river,
For men may come and men may go,
But I go on forever.
So, you could consider this the chorus of "The Brook," but you aren't singing it every other verse. With the chorus of "Yankee Doodle," you are ("A" = verse, "B" = chorus with different melody):
AB AB AB AB . . .
Here's the overall structure of "The Brook," where
"A" represents a 4-line verse with new imagery, and
"B" is a 4-line verse that always has that same theme on lines 2-4
(and again, "B" really has the same "melody" as "A"):
AAB AAB AAB AAAB