The alternative history of the show doesn't diverge from real history during WW2, it diverges much earlier. This is from memory, so I might be a little off, but the gist is:

• President Roosevelt is assassinated soon after entering office. The New Deal never happens and the depression lasts into the 1940s, with American industrial production remaining extremely low.
• FDR's replacement tanks in popularity due to the economic situation, and at the next election is replaced by a strongly traditional isolationist government that refuses to aid the UK or USSR or to blockade Japan.
• The Russians begin Operation Barbarossa earlier in the year and with better anti-tank weapons, leading to the eventual capture of Stalingrad and Moscow and the genocide of all Slavs, with the USSR territory turned into lebensraum. The USSR's biggest oil reserves are discovered and exploited in the 1940s rather than later in the 20th century.
• The USSR's industrial capacity is filled by the Italians, who become better equipped and an actual threatening force in the war, colonising the bulk of Africa and preventing Allied operations in the Mediterranean. With the USSR quickly conquered, what was a two-front war with a brutal Eastern front in our reality becomes a one-front war in which the entire Axis attacks Western Europe, which eventually falls to them.
• The Japanese attack on Pearl Harbour goes ahead, and is more damaging due to the US's lagging industrial capacity. The Americans cannot raise the sunken ships from the shallow waters of the harbour and cannot produce new ships quickly or in significant quantity, so they are dominated navally, and the Japanese navy conquers Hawaii and the Pacific. With the resources gleaned from the new territory and provided by the Germans from the USSR oilfields, Japan conquers China and India. Eurasia now belongs in its entirety to the Axis.
• Now that the USA has little money to spend and an isolationist mentality, while Germany has vast resources and the captured Soviet scientists, the Germans develop nuclear weaponry first. They test it on Washington DC, decapitating the US government, and immediately invade from the east coast while the Japanese invade from the west.
• With the Axis possessing all of Asia, Africa, the Pacific and Europe as well as the USA, it becomes trivial for them to steamroll South America, India, and the Middle East. Independent states that survive, like Canada, survive almost as puppet governments as it's obvious that the Axis is both capable and willing of conquering it brutally should they wish to.