I'd say...United States stays neutral in the First World War, though due to greater German immigration in the 19th Century and backlash against Entente propaganda techniques there is little sympathy towards Britain and France. As such, a sheer war of attrition brings about a negotiated peace in the West with minor border changes, which include - at least initially, the cession of Alsace-Lorraine to France and the payment of reparations in exchange for France's colonial empire and a free hand in the East in drawing up the borders of the now-defunct Russian Empire.
As the war turns against the Russians, revolution breaks out and once the Soviets take power, they sign the treaty of Brest-Litovsk early in 1918. Seeing a wall of nations firmly in the German realm, the Soviet union looks southward, invading in the post-war years. In the 1920s and 30s, Trotsky, perpetuating the doctrine of the Perpetual Revolution, manages to come out victorious against the industrialisation camp which Stalin leads. As such, Soviet forces stream out from the Caucasus into Turkey and Persia. The nascent Iraqi State is thrown into a civil war, where the northern, Kurdish-dominated, half supports integration with the Soviet Union, and the southern state prefers its pending dominion status with the United Kingdom when oil is discovered throughout the desert and British citizens are encouraged to move into Mesopotamia to help 'integrate' the country after 1926.
Due to Ottoman incompetence in the war, they come out the definite losers in any post-war treaty, signing away much of Mesopotamia to Britain and France initially.
The sheer price of the FIrst Great War however was not all rosy for the Austro-Hungarian Empire, and by 1921 they splintered into nearly a half-dozen states. Austria, Hungary, Czechia, Dalmatia, and Yugoslavia all came out of the splintering.
Untouched by the First Great War, the United States found itself in a unique position, funding all of Europe's restoration projects in both Entente and Central Powers nations. However, even this wasn't enough to prevent the Second Great War, beginning in 1932 with the Nipponese invasion of China to 'protect' Nipponese investments in the Manchurian rail system and ensure that the 'oppressed minorities' of Manchuria were fairly treated.
Due to American interests in China, the United States got involved with the sacking of Shanghai and Nanking by overzealous Nipponese forces. Though Nippon apologised for its the massacres and the near complete razing of the two cities, American forces began a westward drive over the Pacific to stop the threat to American interests. Thirteen years of heavy fighting later, and Nippon was integrated as an American-Administered Overseas Territory. Aside from a large naval campaign and the island-hopping, three corps of the US Army as well as a Marine expeditionary force were shipped to China to support the legitimate government, langing in Hainan and marching north to drive against the Nanking pocket. In a long series of joint Sino-American offenseives, by 1942 Beiping was liberated and a counter-invasion was launched of Manchuria which stopped in 1943 only when the Sino-American forces came to the Japanese fortress line along the Yalu River.
The Second Great War in Europe took a different course, as in 1935, dissatisfied Austrians declared war on Hungary to restore their empire after total economic collapse ruined their state. The brief lived Second Dual Monarchy was imposed, and for __ years, all of the Balkans were under Austrian control, as far south as Greece. France and Great Britain sided with the Austro-Hungarians alongside the Italians, hoping that a reformed Austria-Hungary might curb German influence in the Low Countries, Eastern Europe, and the Baltic States. Germany, fearing a large, powerful, and militant nation to her south, allied with the United States to maintain the status quo.
A massive war spanning four distinct European conflict zones was waged, with US forces landing in Ireland to defend it after pinching out British forces from Northern Ireland. An invasion of Scotland and a massive drive south laid waste to the country as the United States conducted a trans-continental bombing campaign with experimental super-high flying delta-wing bombers powered by German-supplied jets. By the time Paris was occupied on October 19, 1943 after triple landings in Calais, Normandy, and in Gascony near Bayonne. The Calais landing was swiftly defeated, and the Normandy landing was bogged down by heavy fighting in Cherbourg and elsewhere along the Breton peninsula. It was teh thrust upward, from Gascony, that was most successful as the 9th Army approached from teh south after liberating, amognst other cities, Toulouse, Bordeaux, Limoges, Potiers, Bourges, Tours, and of course Paris itself. Forces in southern France, cut off from supplies in the north as most forces were diverted westward to engage American forces sent to shore up the flanks of the drive, retreated into Italy, with many finding themselves fighting in Prague and Pressburg.
The Dual-Monarchy's army, consisting largely of non-Austrians supportive of a return by the Habsburgs to the throne, invaded southward in 1937, rolling up Serbia and Albania in ways which they simply could not in the opening strikes of the First Great War. However, the Hellenic Republic, starting in 1933 and fearful more of her Serbian and Albanian neighbours than a resurgent Austria-Hungary, built a line of fortifications along her northern frontiers in every mountain pass large enough to pass an army through. And the armies did come in 1937, right into the guns. For the next two years, Greece became the Austrian's open sore, even as the Austrians slowly pushed down the peninsula, finally taking Athens in late 1941 after four years of very heavy fighting. The mountainous terrain was conducive to a guerrilla war, and particularly in Thessaly and Epirus, where the heaviest fighting occured in the mountains, partisans ensured that the Austrians never did truly control the entire country. For the most part, these Greek partisans were supported via clandestine shipments of Russian arms and advisors sent via Russia. This pressure to open the Bosphorus all but made Turkey into a Russian puppet state, and in emergency elections called in 1942 by the ruling (Right-wing) government in Turkey, the Right-wing was firmly thrown out and replaced by a strong Leftist government that increasingly aligned itself with Russia for protection.
The colonial wars were lively as well, with Italian forces holding their own in Libya against resurgent Senussi, and as such did not participate in the war. In March 1943, realising that the war was turning against it with the capture of Paris just five months ago, Rome signed a separate peace with the Central Allies, switching sides - promising military assistance against the resurgent Austrians in exchange for a preserved colonial empire and assurances of historical territories along the Yugoslav coast. President Roosevelt and Wilhelm III agreed, giving Vittorio Emanuele III the cover to press his government, who eventually agreed later that April.
Allied forces, gathering in Bavaria as well as north of Venice, found the mountainous terrain of western Austria to be quite troublesome, and the fighting was slow and brutal, the street-by-street and building-by-building slog through Vienna took nearly three months as Austrian troops elsewhere in the country were arrayed into ad hoc units to try and relieve the Austrian capital. To no avail however, and on June 19, 1944, the capital of the Habsburg empire fell.
Over the next two months, various disarmament campaigns were conducted, with light fighting in Hungary, Romania, and Yugoslavia as the occupation zones were decided upon. France was divided between East and West, with Alsace-Lorraine returned to Germany. Scotland was given freedom from the United Kingdom, and Cornwall and Wales were devolved, giving them their own parliaments in 1948 after plebiscites were initiated. The Dalmatian coast was given to Italy, per the 1943 treaty, and the Austrian Empire again dismantled. Austria itself was incorporated into the German empire, with American and German politicians alike hoping that the stability of the German state might calm the Austrians, and that the Kaiser in Berlin might be able to fill the apparent need the Austrians seemed to have for a monarch.
In the peace treaties, Nippon was put under complete American occupation. Southern India was likewise given over, where a vocal Judeo-Christian minority, as well as the Tamils, wanted to be under Western rule on account of distrust of Delhi, who was given control of the rest of India. According to the treaty that divided the country, the two states would be set up along religious lines, and India's name was thus chnaged to Hindustan, the northwestern lands where there was a Muslim majority to Moslemstan. East Moslemstan was given to the Germans, who required a naval base in the Indian Ocean, and Burma as well was given to the Germans. Siam, invaded in 1883 by the Italians, was given minor border adjustments as it was formally sandwiched between the German colony and American Indochina.
Tibet, as well came under American occupation when China devolved into civil war and the International Council was beseeched by the Tibetan government for protection as the last remnants of the Qing dynasty collapsed. Communist, Anti-Communist, and Monarchist forces all tried to grab various parts of the country. Tibet, claiming neutrality, required a credible peacekeeping and observation force in the country in order to maintain that neutrality. With the United States in Nippon and Corea already, as well as its garrisons in Hainan (which was de facto American teritory due to the sheer number and size of US military bases in teh area) and Formosa (de jure American, ceded by Japan in its treaty, along with all Pacific Island holdings), it was the logical force to go to. Bhutan likewise plead neutrality, and so the United States set up a small 1500 man garrison there as well. By 1952, China's civil war ended, with an ostensibly republican China hostile to the United States coming out of it at the far end with a figurehead Emperor on the throne. Communists were exiled to the Mongolian frontier where they declared their independence in 1954, coming into the Soviet sphere.
What little was left of the British Empire was soon abolished, with the Congo (sold to Britain by a bankrupt post-First Great War Belgium) and British South Africa going to Germany. South Africa proper was annexed as an overseas territory by the United States as part of the Treaty of Bonn, and the same fate came to Egypt and the Anglo-Egyptian Sudan. Southern Iraq, also formerly British, was divided into two territories - Basra and Baghdad, both of which were given to the United States, along with British Arabia and Kuwait. French Indian Ocean islands, including Madagascar, and both British and French Somalia were allocated as Protectorates by the International Council, to be given independence, along with the other African territories, when they were deemed capable of self-rule. This would not happen until nearly 2000, with the last territory - American Guiana, gaining independence in 1998.
In 1949, the United States put forth the United America Pact to all North American and Caribbean countries. Most joined, and after some diplomatic wrangling, including the establishment of a new currency - the Pan-American Dollar, by 1954, all of North America was under one banner. In an ironic twist, the UAP voted nearly unanimously to take on teh name the United States of America as a very literal description of their status as united (nation) states of the American continent.
In 1953, France was reunited with the condition that American and German observers be put in the French cabinet. As of 1955, the Central Allies still occupy London in a joint command.
Germany, in order to better manage her massive colonial empire, has taken steps towards democratising, and in 1953, under the reign of the curiously liberal Louis Ferdinand. Kaiser Louis Ferdinand even, in 1963, later went on to create the German Commonwealth, devolving all colonies to have a local parliament to decide the majority of domestic matters, with certain domestic and all foreign policy matters still decided from Berlin.
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I think that should explain the majority of this map easily enough.