Complex sentence: For the United Kingdom, the Second World War started in September 1939 and after the brief non-events of the Phony War, the Battle of France saw France and the Low Countries overrun with Nazi German Blitzkrieg.
Split into simpler sentences: For the United Kingdom, the Second World War started in September 1939. After the brief non-events of the Phony War, the Battle of France saw France and the Low Countries quickly overrun by the Nazi German Blitzkrieg.