All I've come up with so far is

On a tree by the river
A little tom-tit
Sang Willow, tit-willow,
Tit-willow.

And I said to him, Dicky-bird,
Why do you sit,
Singing Willow, tit-willow,
Tit-willow...
usw.

But that one's taken.
I'll keep working.