What day did the Germans give up in World War 2?

Simple answer: never.

An air war called the Battle of Britain between 10 July 1940 and 31 October 1940 prevented Operation Sealion from taking place. Germany failed to defeat the RAF and Fighter Command and was forced to scrap the planned invasion of Britain.

Germany attacked England, but did not invade it during WW2. Germany did have plans to invade, if its air offensive had worked. The English name for the German invasion plan was "Operation Sealion".

Germany never invaded the island of Great Britain proper. The closest they got was capturing several of the Channel Islands that are British territory close to the French coast in the English Channel.

The reason they never invaded in 1940-41 was because the Luftwaffe could not destroy the British RAF and gain air superiority that was needed in order to invade. Hitler got tired of waiting and turned his attentions East in June 1941. One factor of the Luftwaffe inability to take control over the skies of England was that the Raf had a fresh supply of pilots from all over the world. Not only the British commonwealth but as far away as Canada, India, and even black pilots from the U.S. while the Luftwaffe pilots had very little relief.