The order would be mostly determined, as usual by what you want to emphasize.
I'd tend to put "flamante" first, because that's the more important idea for me.
Also, I would prefer "something" to go in between both adjectives, as I don't like very much the sound of them together, but if I have to choose, I prefer your proposal --"flamante nuevo".
- Mi flamante coche nuevo.
- Una flamante casa nueva.
- Esta flamante y nueva generación.
- Nuestra nueva y flamante colección de zapatos. (Since it's a product that is renewed often, the most important idea is the fact that it's a new collection, but both orders would work.)
...Yet... if you want to use a "Mexicanism", you may say "nuevecito"... or "nuevecititito", the amount of "ti"s is how much you want to emphasize the fact that the object is brand, really brand new.
- Juan trae un coche nuevecito.
- Tengo el disco nuevo de Depeche Mode: nuevecititito, salió hoy.
- Mi colección de juguetes de la Guerra de las Galaxias (Star Wars), está nuevecita; nunca abrí los empaques.