A reason for not working Budgie autologin is unknown to me. Set thru GUI, it works fine here - auto logins without a problem.
A workaround for Budgie
Edit /etc/lightdm/lightdm.conf file. If it doesn't exist, create it. Put two lines into it:
[Seat:*]
autologin-user=just
just is my user name. Replace it with yours.
To disable autologin, remove the username from autologin-user= line:
[Seat:*]
autologin-user=
Autologin in Plasma
In Solus Plasma it is enabled thru:
System Settings --> Startup and Shutdown --> SDDM Theme (select Breeze) --> Advanced tab --> Automatically log in [ ] as user (checkbox)
The traditional way to enable autologin doesn't work in Solus Plasma:
System Settings --> Account Details --> User Manager --> Log in automatically (checkbox)
A user is asked twice for root password, but enabled checkbox is dropped back to disabled in a few seconds.
Edit: Typo