Well, let's keep this thread going for ever! Mine (South Staffordshire, around 1950) was:

Eeny meeny mackeracka
Rare rye dumeracka
Chickabocker lollipopper
Rum pum PUSH

And if you didn't like the answer when someone was rejected at push you ignored it and went on with:

O U T spells out so out you GO.

And yes, we also used the normal eeny, meeny minie mo (with "if" he hollers, not "when").