I finally succumbed to my curiosity and got the download from Windyridge. As suspected, this is indeed a Harry Champion song and this is my transcription of the track (main differences from Chas 'n' Dave's version are starting with the verse rather than the chorus and no repetition of "Old Bald Head" in the chorus. Oh, and no audience shouting Harem!). I've written out the choruses in full as there are slight variations.

AS LARGE AS LIFE IN THE HAREM
As sung by Harry Champion

Once I thought I’d spend a holiday where the Turkey rhubarb grows.
I put a little bit of rag around my head.
"I'm off to Turkey," to the wife I said.
I had a laugh when I got there, straight I had to do a little grin,
For as soon as the rajah popped from his house, your humble he popped in.

And there was I as large as life, sitting in the middle of the harem.
All the girls got clinging on to me.
Tickling my whiskers, sitting on me knee.
In came the rajah, drew out his sword and said:
"My word! If I catch you bending,
Off'll go your old bald head."

Into the harem once again I went, as soon as ever nighttime fell.
One little lady chased me all around the place,
Dressed in a little bit of flimsy lace.
Out went the gas so I thought, “Wow-wow! Now we're gonna be all right.
There's gonna be a lot of foggy weather at the old crossroads tonight.”

And there was I as large as life, sitting in the middle of the harem.
All the girls got clinging on to me.
Tickling me whiskers, sitting on me knee.
In came the rajah, drew forth his sword and said:
"My word! If I catch you bending,
Off'll come your old bald head.”

I came back to London very soon, thought I'd like to see my wife.
I went in the kitchen and the missus said to me:
"I've got a photograph for you to see.
I've had it sent from Turkey there. You've enjoyed yourself, not half!"
I had a look at it and nearly had a fit, for upon that photograph—

There was I as large as life, sitting in the middle of the harem.
All the girls got clinging on to me.
Tickling me moustache, sitting on me knee.
In came the rajah, drew out his sword and said:
"My word! If I catch you bending,
Off'll go your old bald head.”

(Let’s sing then)

There was I as large as life, sitting in the middle of the harem.
All the girls got clinging on to me.
Tickling me whiskers, sitting on me knee.
In came the rajah, drew out his sword and said:
"My word! If I catch thee bending,
Off'll go your old bald head."

LFF