Matt's idea that the phrase might have a different meaning in each saying seems probable to me.
I've been slightly more successful searching for see a pudding crawl on its own (without the laughing part). And it is a lot more frequent with creep than crawl.
There's a peak in the early 19th century for see a pudding creep because it was used in an essay by Jonathan Swift :
it would vex a dog to see a pudding creep
where it could be understood as "see a pudding go to waste".
The oldest quote I have found is in a 1617 nonsense anonymous verse :
I grant that Rainbowes being lull'd asleep,
Snort like a woodknife in a Lady's eyes;
Which makes her grieve to see a pudding creep,
For Creeping puddings only please the wise.
Here again it's the idea of waste. So how did it become associated with laughter? (A hint of sadism maybe?)
The phrase seems to be British (rather than US), here's an excerpt to what seems to me to be a pastiche from the TV series Star Treck:
"You would laugh to see a pudding crawl..." Spock read aloud, an eyebrow almost rocketing off his face. "...a fascinating image."
Kirk gave him a weird look. "I never thought I'd say this, but that's one of the only phrases I haven't heard Bones say."
"That is hardly surprising, Jim, as it originates in Britain."
It can found in a cockney dictionary and a 2005 excerpt from a web blog:
But then, us Londoners, as my dad used to say, would laugh to see a pudding crawl.
This "blunder" meaning of cock-up has been used before the 1960s, from at least the 1940s in writing.
It can be found in the 1950 Sea slang of the twentieth century: Royal Navy, Merchant Navy, yachtsmen, fishermen, bargemen, canalmen, miscellaneous by Wilfred Granville, which covers the period from 1900 to 1949.
cock-up. A mess-up, a bungled piece of work. A LASH-UP. (Lower-deck.)
lash-up. General confusion caused by a misunderstood order or a bungled job of work. Cf. the Royal Navy's lower-deck term, COCK-UP.
Pierre Clostermann's 1948 Le Grand Cirque is one of the very first post-WWII fighter pilot memoirs:
Hullo Filmstar Leader, sorry old boy, there is a cock-up about the Typhies. Do your best if you can without !
Workers in Stalin's Russia by M. L. Berneri (1944):
The journey of approx. 500 miles took us five days, and has been known to take ten days. As we had to take food for this time we travelled rather like a person moving house. There was a cock-up about transport to take us across the ice to the station
Finally, the term isn't particularly offensive. It's been used scores of times in UK parliament, most recently by Peter Bone:
Last Sunday I attended Indian republic day at the Wellingborough Hindu Association, yet the same week we learn that a £20 billion fighter contract has been lost to, of all people, the French. We now know that the lead bidder was not the British Prime Minister or the British Government, but the Germans. What on earth do they know about cricket and curries? Why was the British Government not leading on that? How did the Secretary of State allow such a cock-up?
Best Answer
Partridge's A Dictionary of Slang and Unconventional English gives this entry:
That entry (interior emphasis my own) leads us to pudding, entry 2:
So it comes from pudding in the sexual, coital sense given above.