It's the 2nd time in a row that people get the complete opposite picture of what I just said. I never claimed that Zuko was back then part of the United Forces, you just turned the causality I was referring upside down, it's the exact opposite I was saying. It doesn't mean necessarily it has something to do with the United Forces, but you can't say either that it doesn't for certain.
Zuko and Iroh are Fire Nation royalty, so it could be explained that it's just at most a fashion statement, but then why would Bumi be running around in it (I seriously doubt it was a mistake in that drawing)? Everyone must have noticed by now the symbolism in the cloths of the people, even in Repbulic City people from the Water Tribes wear blue, Earth Kingdom green, Air Nomad orange and Fire Nation red. Bumi's parents were a water and air bender, so why would he be running around in a fire nation getup? Always possible, but we also know that Zuko tried his entire life to restore the world again and that he had a strong influence on the United Republic of Nations, and thus also on the United Forces (hence is probably the reason as why Iroh is now general).
So saying that the original fire nation fashion made its way as an officers uniform isn't really a far-fetched idea, especially considering such armies are normally not going to have mixed uniforms, which is also backed up by that the metalbending corps consists of people originating from different backgrounds, but wear a common uniform.
On top of that his brother is in the council of Republic City, his uncle was once also in that council and his father was the Avatar. So there would be indeed a very reasonable chance that Bumi started a career in the United Forces and would explain that he is wearing a uniform that originates from the Fire Nation if he would have gotten a high rank in it.
Of course this is nothing more than a possibility, another idea is that Bumi just married Zuko's daughter -as we know he had one- and became Fire Nation royalty. But come one be serious, how is it possible that you people start talking about Zuko being part of the United Forces at that point in his life =/
It could all be fluke to, but as I said I sincerely doubt that's the case.