According to the British Library Sound Archive Catalog, YE CAN COME AND SEE THE BABY was recorded by Will Fyffe in 1927. His recording can be found on an album called "I Belong to Glasgow" (Hallmark, 2008).

Found at Glesga Songs:

YOU CAN COME AND SEE THE BABY
(As sung by Andy Stewart)

Oh, you can come and see the baby if ye care to call.
He's lyin' wi' his mammy in a wee white shawl.
He looks so neat and slanky like a dumpling in a hankie,
And were gonna call him William Angus Jamieson Jock McCall.
Oh dimara doo a doo a day dimara doo mah daddy-o.
We've got a baby in the hoose, and I am his daddy-o.

Well, the wife and I were very, very lonesome for a while,
Sitting in the kitchen by ourselves.
Then one night over tea, the wife she told tae me
A secret that I couldnae keep tae myself;
So at three o'clock this morning, I wis running doon the street.
Oh, the neighbours must have thought there was a fire.
A policeman started laughin', when I came back again,
For by my side was Doctor McIntyre.