Unfortunately, I don't know about the royal families of The Netherlands or Belgium but for Scandinavia I can offer some insight. I believe that the stories in the Benelux is similar to the norwegian, but that is simply based on the danish being regarded as rather unique - in Denmark at least!
While Norway was initially caught by surprise at the start of the invasion in april 1940, a couple of early successes allowed the royal family and the government to escape Oslo before the city fell. They remained in the country at first, but when the british and french troops aiding the norwegians withdrew to fight in France(in early june 1940, if memory serves) the King and government sailed to Britain and established a government-in-exile until the liberation in may, 1945.
In Denmark the situation turned out to be rather special. In the early morning, april 9th 1940, german troops crossed the border, while parachuting men to secure the Aalborg airfield - needed for the invasion of Norway - and other strategic locations, along with multiple amphibious assaults, and airplanes dropping propaganda leaflets asking the danish people not to resist.
While the danish army initially fought back, the government feared the Luftwaffe would bomb Copenhagen. Knowing that there was no real hope for fighting of the german forces, the government, after talks with King Christian X, surrendered within hours. If he even had a chance to escape, I cannot say for sure, but regardless he remained in Denmark. While he hadn't been incredibly popular earlier, he became a rallying figure during the war, often riding through Copenhagen on his horse, sometimes even without the Royal Guard accompanying him. A lot of stories have sprung up about how he acted during the occupation, some are no more tha myth - like him wearing a gold star in support of the jews - but others are built on truth, like his passive-aggresive answer to a birthday letter from Hitler himself.
The situation in Belgium was as with most things Belgian, quite complicated and very controversial. I'll try to give a quick summery.
When the Germans invaded, the Belgian government ordered King Leopold III to flee and go in exile with them. Leopold III however wasn't on good terms with the current government of the time. He saw such a move as cowardly and decided to stay with his troops and continue fighting agains the Germans.
When the situation of the Belgian forces became unattainable at the battle of Dunkirk, Leopold III surrendered to the Germans. He being a constitutional king, doing so without having consulted the government in exile first was scene as an act of treason. Despite being warned of the surrender in advanced, the British weren't too happy about it either as this lead to one of their flanks being exposed.
In the mean time, Leopold III was placed under house arrest in his palace in Laeken by the Nazis. He refused to cooperate with them forcing the Germans to instil a military government instead. After much insistence, Leopold III finally managed to get Hitler to meet with him. He had hoped to convince Hitler to make a statement about the future return of independents of Belgium. When Hitler refused to do so, it at least became publicly clear that the king wasn't collaborating with the Nazis. But while he was still defending his own countrymen, he made no effort to cooperate with the government in exile.
In the later stages of the war, he was deported to Germany and then to Austria. After the war, it was advised he not return to Belgium as his role during the war remained controversial. In Belgium, the south of the country saw him as a traitor, the north saw him as a hero.
A referendum was held and a slight majority voted for his return. But when he did return, riots broke out and it was clear he was no longer representing the unifying symbol of what it is to be the king of the Belgians. To prevent further unrest let alone civil war, he abdicated from the throne and his son became the new king of the Belgians.