Yes, the British probably had the most "active" royal family in Europe but they weren't the only ones by any means.

The German royal family, the Hohenzollerns, had a number of members serving in the German armed forces. Hitler in his early years had oppressed the monarchy's surviving members and their supporters in the hopes of crushing any chance of a monarchist coup, but once that threat disappeared they were allowed to serve in the Wehrmacht. For example the Crown Prince Wilhelm (the grandson of the last kaiser Wihelm II) was a low level commander in the Wehrmacht and served in Poland and France, he was wounded and died in a field hospital on May 26th, 1940 from wounds he got during the fighting in France. There were numerous other German princes who fought in the war.

The Dutch, and Norwegian royal families were not able to do much besides provide moral support, because both families had fled into exile when the Germans invaded. Wilhelmina of the Netherlands and King Hakon of Norway often gave speeches and such in favour of the war effort, but could do little beyond that. This also applies to the Yugoslavian king, Peter II who fled into exile in London.

The Belgians and Danish Royal families were the opposite. Both refused to flee when their countries were invaded. Both spent the war under heavy guard and essentially functioned as mere figure heads without any power.

The Italian royal family was very similar to the British with a number of family members taking up key positions in the military or civilian administration. The Duke Aosta, Prince Amedeo, served in the Italian airforce and saw some minor action against rebels in Libya. He became governor of Italian East Africa and when war broke out he commanded the Italian troops stationed there. He even took personal command of a garrison in the late stages of the campaign. He was captured and died in 1942, in a British POW camp in Kenya. There was also Prince Aimonem, a navy officer who became the King of Croatia. He never enjoyed being King though and he eventually abdicated the throne when Italy surrendered.

There was also the Kings of Bulgaria and Romania. The Romania king was largely a puppet of his Fascist Prime Minister, Ion Antonescu. But the Bulgarian king Boris III played an active role in his government and armed forces. Bulgaria wasn't really a big player in the war, and they didn't really fight against the Soviet Union until the Soviets were right at there borders, so he didn't really lead men into battle. But Boris did govern his country and was responsible for enacting anti-Jewish legislation and participating in the partitioning Greece.

Now if you want to include Asia. The Japanese royal family were active participants in WW2 as well. Prince Naruhiko Higashikuni and prince Yasuhiko Asaka were both involved in the Japanese general staff and Asaka commanded troops in China, including the ones that took Najing. Prince Kan'in Kotohito served as head of the Japanese general staff until 1940. It was customary for every members of the royal family to hold a military commission, only Hirohito's sons were excluded from this.

The Emperor of China was installed as a puppet monarch in Manchukuo (a region in North China controlled by Japan). He wasn't really that influential but did influence politics within China during the war to an extent.