Is "Biggles Flies Undone" a pun?
It is a (rather old-fashioned) pun on 'undone' but it turns on Lexico's third definition of the word:
undone
formal, humorous (of a person) ruined by a disastrous or devastating setback or reverse.
‘I am undone!’
"Alas! I am undone!" is still in common, and histrionic, usage. As a pun it was a favourite of Frankie Howerd's character in Up Pompeii.
So Cleese's pun says Biggles flies (despite being) ruined, and it also says (as you say) his fly is undone.
But...
In the UK we used to call them 'flies'. Plural. Cleese's generation certainly did. The buttons resembled flies, and as there was always more than one of 'em the word was always plural. I daresay it was the advent of the zip that led to the singular 'fly' being used.
So the second sense of the line is not "Biggles's fly is undone" but the more arresting and headline-like "Biggles's flies undone".
I should add that 'I am undone' was not deemed humorous until the twentieth century. "Woe is me! for I am undone;" [Isaiah 6:5, kjv]