Well in the case of Poland most went through "neutral" countries and took ship to friendly places like Britain or France. The Polish government in exile tried to flee after the Germans invaded. They went into neutral Romania who promptly arrested them and gave them to Germany. The Polish government in exile was formed by diplomats and Polish officials already in allied countries. Many Polish military units either went through Romania and Hungary or took the last Polish ships leaving harbour to Britain and France. When Germany attacked the Soviet Union, the Soviets released the Polish prisoners they captured and sent them through the middle east into British hands.

Source:

Third Reich at War: Richard Evans

Second World War: John Keegan