Hitler's advance on Britain, as depicted in the show's intro.
Hitler was the primary (but ultimately unseen) antagonist of the British comedy-series, Dad's Army, although he is based on the real dictator of World War II he can be considered a different entity due to the fact that the series, while being set in World War II Britain, takes some liberties with its setting so as to lighten the otherwise terrifying reality of life during this time period (since the show was a sitcom, which played up the absurdities of its characters and the rare villains they encountered).
As with reality Hitler reigned supreme as the absolute dictator of Nazi Germany and sought to invade Europe and possibly many other regions of the world, obsessed with racial purity and extreme authority his unique take on fascist ideals led to the breakout of a World War and the deaths of countless combatants around the world : he is also infamous for being the inspiration behind the Holocaust and similar atrocities, which further upped the loss of life and made WWII one of the most costly conflicts in terms of civilian casualities, with millions of innocents being murdered by Hitler's regime in what they refered to as the "Final Solution".
Since such a tremendously evil and reviled figure would be unsuitable for a sitcom (not to mention the impossibility of having the supreme dictator of Germany appear personally on British territory) Hitler was never faced in person or encountered within the show but his threat was constant and his name frequently mentioned.
Hitler's brutal blitz of Britain was underway in this series setting and part of the Home Guards' jobs was to constantly watch over their territory for signs of Nazi invasion - with a few Nazis actually appearing in the show itself, though they were suitably absurd and easily dealt with (the most famous of these being the U-Boat Captain).
Hitler would likely continue to terrorize his setting until he suffered the same defeat as his real-world counterpart and committed suicide in his bunker to avoid capture by Allied and Soviet forces, who would proceed to end the Nazi regime once and for all.