Here's a quick transcription of Michael's tune, along with the standard tune (from Mamalisa.com - just the first source; nothing special about it). I've transposed Michael's from Ab to G for comparison. It's interesting to see that in Michael's version the pentatonic original has been further reduced to just 4 tones.

I think Michael might be overstating the case for a missing pause in the German - the two girls duetting on it (linked above) certainly put in pauses (as well as obviously having a great time with it!).

I was wrong in my earlier statement that the song dated from the 1830s. In fact it seems to date from 17C, the first published version being in the 1830s (with just the 1st 3 verses then). The last verse (...people happy in the immediately preceeding translation) seems to date from the 1920s. More information (including the facts that it has been used for childrens games and had political/social interpretations made) can be found in this short article: Populäre und traditionelle Lieder. Historisch-kritisches Liederlexikon.

Mick

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X:1
T:Upon A Tree A Cuckoo
S:Michael Grosvenor Myer
L:1/8
M:2/4
K:G % original in Ab
z2 D2|GA BA|G2 D2
w:Up-on a tree a cuck-oo,
DD/D/ DD|DD D/D/D|G2 z
w:Hi to my rink-tum ruck-sack tick-er man gee,
D|GA BA|GD z2|z2 LGz|]
w:Up-on a tree a cuck-oo sat.

X:2
T:Auf einem Baum ein Kuckuck
L:1/8
M:2/4
K:G
d4|BG Bd|(dc) A2
w:Auf ein-em Baum ein Kuck_uck
GG/G/ GG|GG/G/ BG/G/|D2|
w:Sim-sa-la-dim Bam-ba Sa-la-du Sa-la-dim,
d2|BG Bd|(dc) A2|G2|]
w:auf ein-em Baum ein Kuck_uck saß.