So to remove GDM3
sudo apt purge gdm3
To remove gnome-shell-common
sudo apt purge gnome-shell-common
Remember you can in Ubuntu install multiple desktop environments - you choose which desktop environment you want to launch via the login screen - so click on the “budgie” icon next to the password field to see which desktop environments you want to switch to.
When I say “break GNOME Shell installation” - its just that it means you will no longer have the ability to login to GNOME Shell via the login manager.
Assuming you have the package ubuntu-budgie-desktop installed, this will be enough to ensure future updates & upgrades work OK.