Aaaaannnnd... I found it!

http://www.christysclipart.com/campmidi/theaustrian.mid

This rather chunky piano rendition is the same melody as the one I know for "Peter's Fountain" which I quoted in full above. So - have I helped at all here? Or just given the original questor more to follow up? :)

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Malcolm: I'd be interested to hear the tune for your round. It very much scans like yet another 'oldie from my mother', and yes, this too is a round....

Deep in the shady thicket
The cuckoo singeth clear
Awaken, little shepherd,
The cuckoo singeth clear.

Cuckoo, cuckoo, the cuckoo singeth clear!
Cuckoo, cuckoo, the cuckoo singeth clear!

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so do do do mi do do
mi re do re mi do
so do do do mi do do
mi re do re mi do

so re, so re, so fa mi fa so mi
so re, so re, so fa mi fa so mi