In a world where the Axis won the second world war and the world was turned into something loosely similar to the world of TNO (The axis defeat all of their major opponents with only the exception of America with the war drawing to a stalemate) I think the world would be something like this:
The world would be divided into five spheres or worlds.
The first sphere is the pro American sphere
The second sphere is the German block
The third sphere is the Italian Empire along with it's vassals and friendly nations
The fourth sphere is the Japanese Empire and their East asian co-prosperity sphere
The fifth sphere is more of a wastebasket taxon to describe the rest of the world much like how we use the term third world in our cold war.
These spheres are more or less (at least somewhat attempted) isolated from each other both culturally, domestically political, and potentially physically (in case of physical barriers) from each other. However all spheres interact with the fifth sphere geopolitically and the second and third sphere interacts with each other regularly (after all they share a border). The first sphere does regularly interact with the other spheres but in a more cold fashion. The point is that you would see different developments happening in these individual sphere that would take a considerable force to might break into the other spheres.
Overtime the geopolitical situation would change overtime and a possible winner to this world's cold war might happen. But overall humanity would experience a different but similar change of motions compare to our world.
To expand on this more
The first sphere (United States, Canada, New Zealand, Australia, and most of Latin America and possibly South Africa and Rhodesia here)
- For America this Cold War is about safe guarding American Freedom at all costs while America is now on it's back foot. For America there enemies represent an all entrancing authoritarian racial evil that needs to be destroyed in order to protect it's people for tyranny.
-America doesn't have influence in Europe or the Far East and is effectively on it's backfoot for at least first few decades of this Cold War.
- The narrative in America is that of survivors guilt of not intervening sooner to save Europe from Hitler and now being the sole surviving leader in the free world.
- The first American proxy war would most likely going to be fought in Puerto Rico against Puerto Rican nationalists (like the people who tried to shot up the US capital building in 1954 in otl) armed by the former Axis in an attempt to weaken American holdings there.
-Throughout Latin America, the United States will continue intervening in there. The United States might back leftist death squads and guerillas to counter rightwing movements backed by Europe there. However the United States will back an Augusto Pinochet for an economics advantage.
- Without partners in Asia or Europe the United States might try to economically invest in helping Latin America here more so than in Otl at least in the short term.
- Within America there will be a brown scare somewhat comparable to the red scare of the 1950s. A brown scare did happened in real life during the 1940s but here it will continue. Ironically most the people put on the blacklist and those that protest the red scare in real life will support this brown scare. The brown scare wont receive the same kind of push back the Red scare did, it's blacklist might cut deeper and go after regular normal everyday Americans and will last longer. Senator McCarthy probably wont be seen as the leader of it as well.
-Charles Lindbergh would have his reputation prematurely destroyed with no chance from the US government of rehabilitating that. As the cold war progresses and America get entangled into foreign conflicts, there might be some dissident right wing anti war groups inspired by Lindbergh that could even evolve into a sorta right wing hippie movement.
- Levittown and Levittown inspired suburbs may not exist as they were created as a financial deal for returning veterans when America won the war. Though the housing problems America faced during that time will still occur so perhaps the suburbs that we are used to will eventually still be created.
-The American Left would no doubt still exist
- The civil rights movement might get pushed back a little. Americans in the aftermath of loss from the second world war might not want to care about the plight of black people. Though I do think the civil rights movement will still happen.
You might even get Japanese spies trying to influence the liberal and american left wing. They would be inspired by predecessor Japanese spy Satokata Takashi and the Black Dragon society. We could even see a Japanese spy like a real life Yuri Bezmenov defecting to explain ideological subversion and active measures to American
- I could see Aldlai Stevenson II or Thomas Dewey possibly becoming president during the 1950s
-Who will become president in later decades will be unpredictable but I'm sue we will see some familiar faces.
-American music at first wont change with 50s Rock and Roll still existing but once we get into the 1960s things become unpredictable as the cross cultural impact and infusion of British groups that exists in our world wont here.
So who knows where the music of the anglosphere will go from here.
- I believe that the American people especially back in that time was very strong and resilient and no matter how bad the situation may be will always hold on to freedom. They will overcome past failures and push to victory here.
If and when in the advent of the former axis begins to crumble you will no doubt see american diplomats and possibly boots on the ground to possibly secure and liberate Europe and Asia.
-Canada, New Zealand, Australia, South Africa, and Rhodesia will go through an identity crisis without father Britain but will politically form a sense of cross Anglo brotherhood with each other.
The Second Sphere (The Greater German Reich, The reichskommissariats, Norway, Denmark, Netherlands, Belgium, Britain, Ireland, France, Burgundy, Hungary, Slovakia, Romania, and Bulgaria).
-For Germany this cold war is about spreading and protecting the rebirthed Aryan people against subversive foreign forces.
- Germany will be America's main rival on the world stage and would have to learn quickly not to make any missteps or else the Americans will taken advantage of it.
-It's capital Germania will probably not be located in Berlin as any big building in Neu Berlin would sink into the loose muddy sediment. Germany's New world capital would have to be located in somewhere else.
-Germany and America will wage an economic war of mercantilism. However not a single product from either country will arrive on each other's shores, instead this economic war will be about spreading each other's economic influence on the rest of the world.
-Germany will send a satellite into space in the late 1940s at the request of Werner Von Braun thus beginning the Space Race 10 years earlier. Soon all 4 of the main spheres will have an attempt at entering the Space Race.
-Germany will try to assimilate the Netherlands and Denmark (As Hitler considered these people to once be German) to the point of possibly of coming into conflict with locals including local Fascists.
- Depending on how well relations are Between Italy and Germany go right after the war either France is immediately reunified after the war and the German armies withdrawn from Northern France with the Vichy Government allowed to return to Paris. Or France is split in two or possibly four with the North part of the country in the German Sphere and the Southern part in the Italian Sphere. Along with a possible independent Brittany and SS control Burgundy. The independent Brittany would become part of a growing Celtic cultural revival movement along with the reintroduce Irish language in Ireland as part of an attempt to create a Celtic renaissance by far right Celtic nationalists.
-To the East The Germans are ramping up Generalplan Ost at least to some version of some kind. The Germans can't hide what they are doing to East Europe from the rest of the world forever.
- Almost all of the Jews in Europe are gone and the Slavs are facing the worst event in their entire history.
-The SS themselves will continue to grow and will gain a reputation within German politics similar to like what the Janissaries were in the Ottoman Empire, or the Praetorians in Rome or the Eight Banner Armies were in historical China.
The point is that they will always have a corner in the German Government that the rest of German Government might not want to mess with. There might be a private fiefdom for them in Burgundy or in the Baltics if the situation becomes too sketchy.
- Adolf Hitler retires as Fuhrer not long after the end of the Second World War. Herman Goring becomes his successor however both of them will probably not live too far into this Cold Was and will both die sometime in the Early 1950s.
-Either Heinrich Himmler or Joseph Goebbels will become Fuhrer. If the situation becomes too dicey then the Wehrmacht might intervene and they will impose one of their generals as Fuhrer.
All options will have a drastic outcome on Germany's domestic and foreign policy. Himmler will destroy any good relations with the Japanese and Goebbels will make enemies with the Italians.
-One of the bigger political conflicts within Germany will be over the influence of the Wehrmacht and the SS with only one of them can possibly win over the other with the other being dissolved at the end.
-Overtime different faces of leadership will come and go. You might see the children of Goebbels, Himmler, or Goring becoming Fuhrer and creating political family dynasties. There might be demand that a soldier that fought for Germany in WW2 should be Fuhrer or someone from the Hitler Youth generation. However as the decades pass Germany will become increasingly unrecognizable.