September 1939 to May 1940 on the European Western front always puzzled me. Why did Britain and France just sit tight? Wouldn't it have made more sense to take advantage of their local numerical superiority whole the majority of the Reich's troops were busy in Poland?
In war, the prewar doctrine, training, and war plans are huge. The French had built up a defensive mindset based on the Maginot Line. Switching the mindset and abilities of a million plus man organization is difficult and takes time. The French had a promotion system designed to train and promote Generals who best fought defensive war. The regiments were trained/exercise in the war plan. The NCO mindset is a defensive war plan. As Allied leader, you have a 36 day window to act, before Germany starts to move troops west. For a modern parallel, imagine that you told the US army you would need to fight a light infantry war in Mexico, that the USA could not use any air power (complete doctrine change), and that the war has to be started with at least 500,000 soldiers on the ground, and all this has to be done in 36 days.
Now to the Post fall of Poland window. Germany only had to rail units across Germany/Poland. The UK had to form new units from scratch. So if it did not make sense to attack in September/October 1939, it would not make sense to attack an even stronger German Army in the winter. And after the winter was over, the Germans attacked. Also, due to WW1 legacy issues, the allies could not have attack through Belgium so they have to start the attack crossing the Rhine with green units versus experienced German units.
Now if you are asking why France/UK thought Poland could hold off the Germans indefinitely, I don't fully understand that belief. It is partially wishful thinking, partially overestimation of the quality of Poland's military, and partially the belief the USSR would not attack Poland.
Fair enough. And I suppose no one in their wildest dreams thought that Hitler would blitz France so quickly and effectively.
The plan worked better than even German optimist expected. If the Germans had seriously though France would fall that quickly, there would have been serious Sea Lion planning in November 1939. Likewise, Stalin would likely have taken different actions if he thought France would fall so quickly.