I see a circle with an orange up-arrow icon and clicking it asks me for root password to upgrade KDE Neon. However, nothing really happens. No upgrade. Hovering on the icon tells me: "Upgrade Available. KDE Neon on 18.10".

Is this a known issue/bug? Is there a way to trigger that upgrade from a terminal?

There is no 18.10 version of Neon to upgrade to, so whatever Neon does to disable Ubuntu's upgrade notification is not working.
I am not seeing this on my Neon systems.