To what extent would the Freedomites cooperate with Nazi Germany, and what impact would it have on the war effort? How would such a cooperation affect the war in Europe, if at all? What would this mean for the US war effort in the American front?
The CSA would stay with Britain and France because they are winners. Britain and France would go along with this. The reason for that is that it would provide a useful base of operations for the Entente. Defeat of Germany in the Great War leaves control of the world by Britain and France. Well, sort of control. The difference between OL and TL191 is that the USA is not a sleeping giant but a hostile power and one that obviously needs to be contained. Whether or not the parts of Canada that were captured are returned, the fact that the USA were more successful on the Northern Front would be worrying for the British. Thus they would have to create a larger garrison.
In the south in the next war the CSA will obviously need reinforcing, probably French troops as the British are committed to Canada. That can not be that many as France is still committed to containing Germany as well.
In the case of the other alliance, the balance of power will have changed. My understanding from start of The Great War, American Front is that it is the USA that is looking for an ally, not Germany. That would make the latter the senior partner.
After a Great War with Germany defeated power switches to the USA for two reasons. One, she won on her front. Two, it is now not a case of getting Britain and France off the back of the USA, but off the back of Germany. The USA can do things such as build barrels and fighter planes that Germany would not be permitted to do so. That give her seniority.
In WW2, TL191 it can be assumed that Featherstone will start a war. However, France may not do so because in this PoD she had no incentive to start one. Instead she will remain behind the Maginot Line. She could still plead a risk of Germany and so not send many troops. This would be even more so if it were timed with the Nazis starting a war.
In a non Nazi war Britain has a choice. Join an aggressor and strike the USA or sit this one out. Given no threat from Germany my guess is a repeat of the Second War between the States, ie enough effort to stop the CSA getting completely flattened but no more. Containment of the USA is easiest done by an intact CSA.
In a Nazi ,to use local jargon the CSA will be sold down the river if vis Canada that gets a neutral USA.