On 1st September 1939, a million Nazi troops then invaded Poland. The United Kingdom and France had promised to help the people from Poland if they were attacked so gave Hitler an ultimatum by threatening war if his troops did not withdraw by 3rd September. Hitler ignored this threat and then the war was declared.
The Germans invaded Denmark and Norway on April 1940. Luxemburg, the Netherlands and Belgium were both invaded in may and then conquered France by June. The Allies tried to disrupt them but then they failed. Chamberlain,(the president of the UK at that time) was replaced by Winston Churchill as Prime Minister of the United Kingdom in May 1940, because the British were not happy with his work.