"The strangest rigmarole came from North Hinksey, on the outskirts of Oxford, in 1985: 'Itchy me, star shee, Logo hutsy yutsy. - kill it', and the game was called 'Itchy me'." (From Children's Games With Things, Iona & Peter Opie, 1997 (Oxford University Press), quoted above)

I went to North Hinksey primary school in 1985 and I remember that rhyme (we called the game "elastics" and also used the "England, Ireland, Scotland, Wales..." one). I learnt a few words of Japanese a while ago and the numbers 1-10 struck me as familiar: Ichi, ni, San, Shi, Go, Roku, Shichi, Hachi, Kyuu, Juu - not quite the same but sounds like a corruption, particularly the first part. There weren't any Japanese children at the school when I was there. I was surprised the rhyme was so local. I can't find any other references to it on the internet.