'Babans' isn't a word. I guess you mean 'to go bananas' which means 'to go crazy.' ' It's a quite outdated expression nowadays. Both will be understood. I wouldn't say they're especially popular.

They are both very well known in both countries. We use piece of cake more than go bananas . Many idioms are understood on both sides of the Atlantic, so it is not something to worry about

It is amusing that there are two idioms related to dessert which both mean that something is easy. The first idiom is the one you are asking about, "a piece of cake." The second is "as easy as pie." Example: "For some people, math is a piece of cake." And: "For some people, math is easy as pie." Both idioms mean the same thing: For some people, math is easy.

if you mean "go bananas" it actually means "go crazy" "he went bananas" "he lost his marbles" "he lost his mind" "he went crazy" all mean about the same

Do you mean "go bananas"?