The Spike Milligan TV version was pretty true to the original, but I think Spike included some extras not penned by JB Morton. In particular the spoof adverts for "Snibbo" and " Threadgold's Thoroughgrip Garterettes" were accompanied by jingles. I'm not sure if the verses for these ads were by Morton or Milligan. Example:

A peer was stalking a roebuck
In a misty Scottish glen.
BANG! went his well-charged rifle.
A miss, then BANG! again.
"TWO misses", roared his lordship,
as his stockings trailed in the wet,
"Serves me right for not wearing
a Thoroughgrip Garterette"

Beachcomber often read from letters he received. My favourite:
"Get your orang-utan off my lawn, or I'll set fire to your bidet"
Signed, Well Wisher

Beachcomber's reply:
"...I know all about you and a certain Miss Maureen Body of Hampstead Heath - and other open spaces"

THe Goons and Monty Python simply picked up the baton.