I've upgraded my devices to iOS 13 and macOS Catalina. When my iPhone prompted me to upgrade reminders, I went ahead with it since all my devices were now new-reminders capable.
Reminders are working on my mobile devices (iPhone and iPad), but on my macOS Catalina device I just get two reminders: "Where are my reminders?" and "The creator of this list has upgraded these reminders." Since I've already upgraded the device to Catalina, I'm not sure what to do to get it syncing again.
The one thing that's odd about my setup is that the iCloud account with the reminders is a secondary account on the device. I don't use reminders with the primary iCloud account on this device.