For those not familiar with the Grasshopper Lies Heavy, it is the title of a story-within-a-story (specifically, a novel) that plays a prominent part in the Man in the High Castle. It depicts a world where the Allies win the war, but in a different way than what happened in OTL. In full, this is how Wikipedia describes the scenario postulated in the story.
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The Grasshopper Lies Heavy postulates that President Roosevelt survives an assassination attempt but forgoes re-election in 1940,. The next president, Rexford Tugwell, removes the Pacific from Pearl Harbor, Hawaii, saving it from Japanese attack, which ensures that the US enters the conflict a well-equipped naval power. The United Kingdom retains most of its military-industrial strength, contributing more to the Allied war effort, leading to Rommel's defeat in
North Africa; the British advance through the
Caucasus to fight alongside the Soviets to victory in the Battle of Stalingrad, Italy and Hungary renege on their membership in the Axis Powers and betray them; British tanks and the Red Army jointly conquer Berlin; at the end of the war, the Nazi leaders—including Adolf Hitler—are tried for their war crimes.
After the war, Winston Churchill remains the British Prime Minister, and, because of its military-industrial might, the British Empire does not collapse. The Soviet Union, crippled by war losses, collapses. The US establishes strong business relations with Chiang kai-Shek's right-wing regime in China after vanquishing the Communist Mao Zedong in the
Chinese Civil War. The British Empire becomes more racist and expansionist post-war, while the US outlaws
Jim Crow laws, resolving its racism by the 1950s. Both changes provoke racial-cultural tensions between the US and the UK, leading them to a Cold War for global
hegemony between their two vaguely liberal, democratic, capitalist societies. Although the end of the novel is never depicted in the text, one character claims the book ends with the British Empire eventually defeating the US, becoming the sole world superpower",
So what would need to happen to make the events of the fictional novel play out in the real world? You are free to make deviations from the story for plausibility's sake.