Well, we'd need a POD that's slightly different from OTL. The British Union of
Fascists would need the right conditions. Like what? A bad economy. Got in spades in the 1930s. A weak government. What would be a good example of that? There are two analogues for that: The Weimar Republic, and Italy, IOTL.
The Weimar Republic was so weak that merest demonstration would bring down the government and the German public was really very apathetic toward
it. They wanted a strong leader and Hitler was that strong leader in Germany.
In Italy, Italian democracy was so corrupt that the Fascists seized power by marching on Rome. Plus, King Vittorio Emmanuel II was forced to name Mussolini his prime minister. Now let's transfer both scenarios to Britain.
Better yet, let's add Austria to mix. In Austria, the Austrian Nazis, assassinated Engelbert Dollfuss, the Austrian chancellor at the time. Let's
suppose Oswald Mosley's British Union of Fascists ran, stood, as the British say, for election in 1934. Mosley says that if he doesn't win, he'll lead a march on London. Meanwhile, his bully boys rough up their opponents, the Tories, Labour, and the Liberals, and the Communists, and blame it on the Communists. Now Mosley knows he can't win the election outright. So he
instigates riots against the Jews, the Catholics, and other minorities. The
King is reluctant to name him as Prime Minister. However, a conniving politician reassures the King that he can control Mosley. The King agrees and
names Mosley Prime Minister. Mosley's first act as Prime Minister is to name
his own Grand Fascist Council, his second act is to round up all his opponents, including Neville Chamberlain and Winston Churchill, and throws them into concentration camps.