My minecraft username is changed but it doesn't appear in-game
Solution 1:
Log in and out of the launcher itself, and your name should update.
Solution 2:
Your profile name isn't your in-game name. You can see that your name is changed by looking to the right bottom of the launcher:
Solution 3:
Mojang's systems are cached. Try closing your game client and opening it up again after 10 minutes.
If this does not work, then some weird error has gone on in the backend, and it'd be logical to contact Mojang.
You can check what the Mojang API is showing in a GUI on www.namemc.com