We saw that in "The Blind Bandit" the Fire Nation has a national anthem that was sung partially by Fire Nation Man. If the other nations actually had one what do you think would be the name, lyrics and what would it sound like in terms of beat? And remember that each of the three water tribes would have a different one.
For some reason I feel like the Fire Nation anthem would be very russian in feel, something along the lines of "Farewell to a Slavic Woman", very strong and triumphant.
The Earth Kingdom, given its rather fragmented nature, wouldn't have just one anthem. There might be one centered around Ba Sing Se, along with several "unofficial" anthems tied to the various semi-soverign land holdings across the Earth Kingdom. I could see the anthem being having very powerful language, meant to reasure citizens even in the darkest of times, along with some very low brassy instrumental background music.
I don't see the Water Tribes really having a national anthem, given the fact that they are even more fragmented and tribal than the Earth Kingdom. They may have a few favorite folk tunes meant for singing around a fire with a hot bowl of sea prunes, but nothing to really give them a more centralized identity.
The Air Nation/Nomad/Acolytes (I don't know what they collectively call themselves these days) would probably be very slow and melodic, evoking the immense sorrow and strife that the element of Air has seen, while at the same time just adding in a few positive chords to give hope to the listener.
The United Republic of Nation's anthem would probably seem very, well, American to our ears, exemplifying the many opportunities to be found in the new nation, the strength in diversity to be found in its borders, and the knowledge that the URN is a shining symbol to all the world that all the nations can indeed exist in harmony.
Specially if FN was not based upon east asian cultures and principles, something Slavic/Russian sounding would be interesting. personally I would like some pre-columbian native american chanting, drumming and panpipes,etc you know? something Aztec for example!! just to think 'outside' the box! ;D
If we're explicitly talking about RL-based, it would be mostly like the Qing Empire anthem, link below. And its lyrics are very FN, putting focus onto the 'perpetuality' of FN rule and eternity,etc.
Considering the circumstances under which we heard part of the Fire Nation Anthem, it probably wasn't the real deal. The guy singing it was supposed to be a heel (his job was to be the buffoon, and hated by the audience). If the Fire Nation has an anthem, it's probably something a little more classy than "Fire Lord, my flame burns for thee..."