I see the following.

Budgie is GNOME based, so should act similar to GNOME. Like we it or not.

Both GNOME and Budgie uses XKB and Ibus; the latter is needed only for complex alphabets.

With default configuration Budgie should read /etc/default/keyboard (see example below)

cat << EOF | sudo tee /etc/default/keyboard 

# KEYBOARD CONFIGURATION FILE



# Consult the keyboard(5) manual page.



XKBMODEL="pc105"

XKBLAYOUT="us,ru"

XKBVARIANT=","

XKBOPTIONS="grp:alt_shift_toggle,grp_led:scroll"



BACKSPACE="guess"

EOF

and set keyboard layouts and their switching shortcut from it. Thus if you have grp:alt_shift_toggle (Alt+Shift) defined here above, it should work.

To add necessary keyboard layouts you have to launch Keyboard settings (gnome-control-center keyboard) and add needed languages to Input Sources here.

If you want to set Alt+Shift or Ctrl+Shift for first time you'll need to install GNOME Tweaks by sudo apt-get install gnome-tweaks --no-install-recommends and set such shortcuts from it in Keyboard & Mouse → Additional Layout Options → Switching to another layout.

Then you'll need to add relevant applet to the Budgie Panel:

1. Launch Budgie Desktop Settings (budgie-desktop-settings)


2. Go to Top Panel tab


3. Click on + button to add Keyboard Layout applet


4. Optionally move Keyboard Layout to the needed part of Budgie Panel


5. Test keyboard layout change and make sure it changes on Budgie Top Panel.

Details are available at https://getsol.us/articles/configuration/configuring-keyboard-layout/en/ .

Note: if your issues persists this maybe permissions issue, you have to reset them by sudo chown -R $USER:$USER ~/.cache ~/.config ~/.local then reboot and retry above steps.

^{While writing this text I feel that both GNOME and Budgie are not viable for real life usage. It is good that fully featured alternatives like KDE, MATE and Xfce exists.}