Yeah Toni has it, its called a button stick. This one is my Dads from when he was in the Royal Navy, I used it when i was in the army. You slide it under your brass button and polish accordingly. Our buttons on our tunics as they were called were removable though through little rings like keyrings.
Thats something i could never understand about the military from a cost point of view. Given when i did what was called national service (conscription) we were heavily involved in a bush war which where most ended up, why kit each recruit out with full dress uniform as well as normal gear. I think this is another hang over from European Military culture that modern armies could well do with getting rid of. Boy they would not want me around as a cost saving accountant. lol
We were given three forms of head dress and only used on. In Sa we had a steel helmet which had a removable plastic inner that was used in basic training without the steel part. Then a beret and insignia for dress and then a bushhat which is all we really used. The steel helmet was only good as a cooking pot if you removed the inner. Of course they modernised it all since then. Same goes for footwear. Smart really good quality leather shoes both upper and soles. Boots 2 pairs rubber sole leather upper, and then utterly useless 1920's type tennis type shoes.
I guess Military waste is a subject best left alone.