It's fairly old, and known by a number of titles. Found mainly in Scotland, it appears, but also in the North of England. There are several texts in the Digital Tradition database:  Robin Redbriest's Testament.  They don't include the fragement you quoted, though; here is a set from Northumberland which does:


THE ROBIN'S LAST WILL

(From the Mitford family, Northumberland. M. H. Mason, Nursery Rhymes and Country Songs, 1909)

As I cam' past by Garrick,
And by the brig o' Dee,
I saw a little robin,
Sitting on a tree.
Tooraloo, tooraloo,
Tooraloora tooraloo.

I said, "My pretty robin,
Hoo lang hae ye sat here?"
He said, "I've lived upon this tree
These four and twenty year."
Tooraloo &c.

"I'm going to mak' my testament
Just here upon this tree,
I'm going to mak' my testament
This day before I dee."
Tooraloo &c.

"I'll give my pretty head,
It is baith round and sma',
Unto the boys o' Garrick,
To play at the foot-ba'."
Tooraloo &c.

I'll give my pretty legs,
They are baith slim and ta',
Unto the brig o' Garrick:
I hear it's going to fa'."
Tooraloo &c.

As little Robin ended,
He shut his pretty eyes,
And doon he dropped unto the groond,
Never more to rise.
Tooraloo, tooraloo,
Tooraloora tooraloo.


Number 3900 in the Roud Folk Song Index.