I have a version with fairly clear lyrics. I'm certain of the following:

"Russia knows her onions when tomatoes fly" and "Doesn't seem to matter that hate mushrooms fill the sky"

("America's hot potato" is nuclear capability. To "know your onions" means to know what's what. "Hate Mushrooms" are mushroom-shaped nuclear explosions.)

I also think that it's "Nobody hit the grouse" (not "had") - a grouse is a game bird killed for both food and sport.