I have finally managed to solve this issue myself, so hopefully, it will help others with similar issues.

If your VAIO function keys or settings in VAIO Control Center are not working as they should if you open up the Services via Computer Management and look for a Service called "VAIO Event Service".

Make sure it is set to "Automatic" and that it is "Started".

On mine, it was disabled and stopped and as soon as I started it, the function keys and keyboard backlight settings started to work again.