My Granny's version was similar and went as follows. Granny was born 1899 in Shropshire and sang this to me in the 1960's, but my Mum seems to remember her singing it to her also, so that would be late 30's early 40's. . .

In the sunset glow by a bungalow on a South Australian farm
Were a dear little boy and a girlie too,
Playing with a big fat kangaroo
In the evening tide.

From the trees a voice calls out,
'You little ones it's time for bed'
So two little tired and sleepy heads
to the kangaroo then said...

Go away Mr Kangaroo,
We're tired of our games and you,
We're off to our beds to cover up our heads,
Goodnight Mr Kangaroo

Now the evening moon comes a peeping, soon and Mr Kangaroo with pride
Creeps close to the door of the bungalow
And he wouldn't go away, oh no oh no,
From his playmate's side.

From the window up above,
Two little heads are popped outside,
Call round again when the morning comes
Mr Kangaroo they cried …

Go away Mr Kangaroo,
We're tired of our games and you,
We're off to our beds to cover up our heads,
Goodnight Mr Kangaroo