Completely remove Budgie

ubuntu-budgie-desktop is a meta package.

A desktop meta package is a master package that installs all related packages for a particular desktop environment.

In this case - all the packages related to Ubuntu Budgie.

So in addition to ubuntu-budgie-desktop you must also purge budgie-desktop. This is the graphical environment itself and it will also remove the login session for budgie.

sudo apt purge ubuntu-budgie-desktop budgie-desktop

There are a few recommended packages also that are installed with ubuntu-budgie-desktop that will not be removed with the above command - but you can deal with those separately if you so wish.

• Uninstall packages:

  sudo apt remove budgie-desktop-environment budgie-desktop && sudo apt autoremove
  

• Remove PPA from 'System Settings' -> 'Software & Updates' -> 'Other Software'
• Revert to default login screen:

  sudo apt remove lightdm-gtk-greeter && sudo apt autoremove
  

• Restore default splash screen (choose 'ubuntu-logo'):

  sudo update-alternatives --config default.plymouth
  

• Finalize plymouth with:

  sudo update-initramfs -u