A further version, similar to the 'Sambo' ones above. This was printed in a mimeographed mountaineering song book that my father has - mid-1950s, probably British. The theme of arbitrary fatal mishaps perhaps appealed to the climbing mindset:
Sambo had an Aunty, an Aunty very poor.
One day she said to Sambo, "I'll have you scrub my floor."
Sambo didn't like it, and he went upstairs to bed,
Came sliding down the banisters and landed on his head,
More work for the undertaker
Another little job for the tombstone maker
In our local cemetery,
Ibe a tombstone you will see,
Sambo had an Uncle, an Uncle very rich.
One day he said to Sambo, "I'll give you two and six."
Sambo feeling thirsty, went into a lemonade shop,
Bought ten lemonades and ten ginger beers and Sambo went off pop!
Sambo had a sister, a sister very thin.
One day she said to Sambo, "I'll prick you with a pin."
Sambo didn't like it, 'cause he was very fat,
And when she pricked him with a pin he went off bang like that.
Sambo went to the railway station, without a hat or coat.
He tried to slide on the railway line on a moldy bar of soap.
'Long came a run-away engine, right in Sambo's track,
And would you believe it, he rolled up his sleeves and he pushed that engine back,