Can you do the following and see if the bootloader shows on startup:
sudo clr-boot-manager set-timeout 5 && sudo clr-boot-manager update
(from: https://getsol.us/articles/troubleshooting/boot-management/en/)
I assume all are installed in UEFI mode, so clr-boot-manager must be the boot manager used by Plasma/Budgie. This should present itself on boot when spacebar is hold down. However, this command should make the boot manager appear also by default.
(The default timeout is 0, this changes it to 5, show it should be visible for 5 seconds)
Edit: Never mind. I see that budgie boot has a different efi partition (/dev/sda2). Probably windows boot is in that one as well. Hence it is expected that Plasma (/dev/sdb2) cannot find windows/Budgie installations, as only the current disk is searched (in short, you will have to use the bios menu to switch between Budgie/Windows and Plasma).
Try to add Budgie to the bios boot menu by doing:
sudo efibootmgr -c -d /dev/sda -p 2 -l "\EFI\systemd\systemd-bootx64.efi" -L "Budgie Boot"