@rogercooper Well, looking at the amount of upvotes for you and @Frostion, it looks like that most people are not bothered at all about the fact that the played element and the playing actor are both indistinctively called "player", that I instead feel extremely confusing and near to impossible to talk about.
Even tho it is not my preferred pick, I would surely take "faction" or even "polity" over "player".
In the basic games there is no Germany or Britain, there are Germans and British. Of course, the British are not only British, but also Indians etc.. In the moment the name is referring to something specific, namely to the people of a particular "country" or "power" constituting the core of that particular faction, I don't see a problem with being coherent with that.
Side note, also "empire" would be a good pick for WWII. British, Germans, Japanese were empires (tho, for the Germans, "Reich" doesn't clearly translate as such; that would be "Kaiserreich"), while Russians and Americans are quite often informally referred as such ("imperialism" and all that), albeit "sphere" is probably more suitable a term.