There would be a monarch (distance family or well respected person/family to take the place of Wu if need be)
I think you miss read the bit about the political nations of course there political I mean they are ran differently to each other in custom and law.
It would be boring with two republics in the world of Avatar but it would be nicer to have the five nations running on different political grounds. And to prove their not all run by monarch here is how their government works.
Water Tribes are run by a chief which helps advise and make decisions for its people. A chief (irl) can be challenge by another and replaced but usually the sucession passes through their kin. The Water Tribes being cold have to hunt for food and reasources and struggle to survive therefor being more uncivilised.
Air Nomads are ruled by a council of five elders that protect, advise and safe quard the customs and laws of the air people these elders are usually picked from the wises of the Air Nomads and replaced after death by another chosen wise Nomad.
Fire Nation is ruled by a Firelord. As we know Firelords are usually violent and into war. Usually a Lord is given land and peasants by a king or higher ruler and told to protect and train their land and army. Ther Firelords match this description but they don't serve a higher person. Firelords are susseded by their children.
Earth kingdom is ruled by the earth monarch which are usually more into the well being of their people (Wu) or money (Hou-Ting) unlike the Firelords they are not really war tested and out their trust into the five generals to command their armies. The earth monarchy is all about high class, style and luxury. There all about grand expensives like a normally rolay family and their sons and daughters take their place on the throne after their deaths.
The United Republic of Nations has a voted in president who runs the country and can only serve for a small period of time.