After the 1940 'fall of France' everyone expected that a peace-treaty would be signed between England and Germany.

While Germany anticipated that Britain would surrender following the fall of France, nobody else did. There was an element of the British public that wanted peace, but most were prepared to fight. In the government, outside of Lord Halifax, the majority of Churchill's cabinet was prepared to fight.

Churchill himself delivered his famous Finest Hour speech on June 18th, 7 days before the Armistice was signed. If you aren't familiar with the text of the speech, here is the salient portion:

What General Weygand has called the Battle of France is over ... the Battle of Britain is about to begin.

Britain was prepared to fight and Britain's preparedness to fight was truly disruptive to Germany's plans. Britain doggedly fought the Germans in North Africa, tying up a quarter million men and funneling valuable equipment and resources away from the East. Britain served as a base for US bombing operations and, later, as a staging ground for D-Day. Britain was a convoy hub, ferrying goods around the Cape of Good Hope to the Pacific. It was vital in keeping the PQ / Arctic Convoy system alive which delivered supplies desperately needed by the Soviets. Britain's continued belligerency tied up soldiers and resources in France which could have otherwise been committed to the East.

And, while you seem dismissive of the Battle of Britain, it certainly had an impact. It cost the Germans dearly in terms of skilled pilots and aircraft and, as the Eastern Front wore on, those pilots would have been very useful. Germany produced fewer than 8,000 aircraft in 1940 and lost nearly a fifth of that number over Britain.

It's impossible to say how the war would have unfolded had Britain fallen or surrendered in 1940, but the Battle of Britain was important. It kept the British, their air force, and their navy in the war. It gave the US a base of operations in Europe. It kept an economy in the war that churned out millions of soldiers and millions of tons of weapons.