I’m having the same issue. All the setup tutorials I’ve watched are for iOS 12. They say to have location services for the weather app set to Always. Then, you must have Bedtime and Do Not Disturb enabled and scheduled. The live weather on the lock screen is then supposed to appear when the DND turns off. It only appears once, right when the DND ends, and goes away when you dismiss it or open your phone.
I have iPhone X running iOS 13.1.3 and I cannot get it to work. I have my Bedtime and DND scheduled, but what I noticed is the Weather app under Location Services (Settings/Privacy/Location Services/Weather) no longer has the option for Always. Mine only has Never, Ask Next Time, and While Using the App.
My guess is that’s why it no longer works. Again, I’m just guessing so if anyone has a solution or clarification for where I’m messing up, it’d be greatly appreciated. It’s a nice feature to have.