In the BBC drama SS-GB the Nazis have invaded the UK and instigated their rule across the country. The timeline of the first episode appears to happen in 1941.
What event was different from our timeline to allow the Nazi invasion to happen?
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Operation Sea Lion (German: Unternehmen Seelöwe) was Nazi Germany's code name for a provisionally proposed invasion of the United Kingdom during the Battle of Britain in the Second World War. Following the Fall of France, the Nazis expected the British to seek a peace agreement, and invasion was considered as a last resort only if all other options failed. As a precondition, the operation would have required both air and naval superiority over the English Channel and proposed landing sites, neither of which the Germans ever achieved during the war. A large number of ill-suited barges were gathered together, but Sea Lion was postponed indefinitely on 17 September 1940 and never carried out. The German High Command had little confidence in the plan's feasibility, and most historians agree it had little possibility of success.
Whereas in the book, it was successful:
SS-GB is set less than a year after Britain’s surrender following a successful Operation Sea Lion. In 1940, the Germans landed near Ashford, Kent, and Canterbury was declared an open city. The German advance captured London but a British rear guard around Colchester slowed down the Germans for long enough to enable Royal Navy ships to escape from Harwich. King George VI and Prime Minister Winston Churchill became prisoners of the Germans. Britain’s gold and foreign reserves were shipped to Canada.
The Wikipedia article for the show notes something similar:
It is set in a 1941 alternative timeline in which Nazi Germany, having won the Battle of Britain, is now a harsh occupying force in the United Kingdom.