I'm fascinated by this. In my native Russian, there's a children's counting rhyme "ene bene raba kvinter finter zhaba" (and the more popular "eniki beniki were eating vareniki").
You can see some resemblence of "ene bene" to "eeny meeny...", and "kvinter finter" to "tethera, methera".
I'm mostly puzzled how it was spread. How did children in Russia learn about this? It's not like it was very common in Russia to see someone from Britain, especially in the countryside.
There's a theory that "eeny meeny" just means "one two" in some original language when humanity spread, and it was conserved in children rhymes, that get transmitted mostly unchanged from generation to generation.
Your Russian version is very interesting, because I also see similarities between "kvinter/finter" and latin "qinque", greek "penta" (both mean "five"), and "zhaba" may relate to Latin "sex", English "six" and so on. So the Russian version may remember even more original number words. The German version "ene mene muh" is sadly degraded compared to the others.
You can see some resemblence of "ene bene" to "eeny meeny...", and "kvinter finter" to "tethera, methera".
I'm mostly puzzled how it was spread. How did children in Russia learn about this? It's not like it was very common in Russia to see someone from Britain, especially in the countryside.