Why do custom global hotkeys not work in Winamp when playing LoL?
I don't know. It didn't work for VLC either. I would suspect, that League of Legends is swallowing those keyboard commands and not letting other applications know about them (there is probably someone who can give a much more technical answer about keyboard hooks etc).
How do I fix this?
Well this I have an answer to! If you go to Options -> Preferences -> Global Hotkeys you will notice an "Enable default multimedia key support" (Winamp v5.623). Check this box. This will let Winamp "hear" any multimedia keys being pressed. Now while playing league of legends you can use the multimedia keys on your keyboard to change songs, volume etc.