The "white guy blacked up with pantomime accent" was the actor Michael Bates. He was born in Jhansi, was a Gurkha officer and served with the Chindits in Burma. He spoke fluent Urdu.
There were a number of Asian performers in It 'Aint Half Hot Mum but Micheal Bates was the only one born in India.
Well there you go. Every day a school day. Still didn't make it funny in my eyes though.
But hey, chacun a son gout and all that. I found 'Allo 'Allo and Dad's Army hilarious, but IAHHM left me cold.