I'm very late to the party here, but I remember a different variation to the Rule Britannia one. I was raised in Yorkshire so maybe in the 80s, pre Internet, colloquial preferences are what played a part here.
But it went:
Rule Britannia,
Three monkeys on a stick.
One fell down and paralysed his
Prickles grow on bushes,
Prickles grow on trees,
Prickles grow on Chinamen,
Just above their knees.
Sally had a baby
She called it Sonny Jim,
She took it to the toilet
To see if it could swim.
It swam to the bottom,
