I found this, but apart from the word "Auntie" there's no hint in the lyrics that it's about the BBC, or that the praise is meant ironically. Another puzzling thing is that none of the performers nor the songwriter is British; they are all European, but they sang in English. The timing is apt, however; the BBC was founded in 1922.

AUNTIE
Words & music by Hans van Hemert, 1972.

1. Here's a chance to demonstrate our gratitude to you 
On your birthday, on your birthday.
Give a bit of credit where a lot of credit's due 
On your birthday, on your birthday.

CHORUS: We're grown-up people now 
But still we say 
It's you who showed us how 
And led the way 
And so we thank you, "Auntie" of renown, 
And let you know that we shall never let you down.

2. Who's the family relative to whom we owe the most? 
It's our "Auntie", dearest "Auntie."
Ev'rybody rise and raise your glasses in a toast: 
Here's to "Auntie", here's to "Auntie." 

3. History was in the making when you came along 
On your birthday, on your birthday. 
Fifty years of age today and still you're going strong 
On your birthday, on your birthday.

4. Even though you're fifty years old, your kids still can see 
You were great, you were great. 
So congratulations, "Auntie", all the best, and "Cheers!" 
And we all say "Happy Birthday."

[Recorded by Vicky Leandros, Demis Roussos, Enrico Macias, Sandra & Andres, Hildegard Knef, and Alice Babs (Apparently all these people sang on one recording: Philips 6832032.)]