A very similar question from a couple months ago has some great answers to your question.

The Axis actually named themselves the Axis though they left off the evil part. The term came from the Pact of Steel between Germany and Italy. Most of the time though they just referred to themselves by individual country.

The allies were referred to as Bolsheviks or Plutocratic-Capitalists. Often these terms would be combined with a phrase tying the enemy to a Jewish conspiracy.

The average soldier on the ground would refer to an American soldier as a yank or an ami Russians were Ivan Brits were Tommy.

Your assertion that the Allies used the term "Axis of Evil" is not supported by facts.

A Google Ngram search of the terms "Rome - Berlin axis" "axis powers" and "axis of evil" shows that the first two terms were widely used from 1935-45, but that "axis of evil" was essentially unheard-of in English language media until 1975.

Edit- A search of French written media also indicates that les puissances de l'Axe was/is the preferred term, and Axe du mal was similarly unheard of.

Small note, some of the "Allies" preferred the term "United Nations" at the time, and to avoid confusion with the present day "United Nations" that term has fallen out of usage.

FDR called the "Allies" the "United Nations" (he actually came up with the latter term)