There is no direct translation for "Ally" and "Alliance". Often we use the words "союз" and "союзник". But you should use it right, because it is more "union", a stronger word.

For alliances we also use "военный союз/альянс" and for allies "союзники по альянсу".

For Allies and Axis there are fixed notion "Страны Союзники" and "Страны Оси".