@scottkendall just to clarify your answer, after updating to 10.15.3 you reinstalled macOS 10.15.3 on top of the existing installation and that caused unlock with apple watch to start working again? Can you also confirm you had the same exact symptoms where all other handoff/continuity features work including approve with Apple Watch (approve admin dialogs by double tapping the watch button) but unlock with Apple Watch shows a spinner at the login prompt and them times out and asks for the password?
I haven't had it working in my Catalina system since upgrading from Mojave. It seems weird that an in-place install of macOS would fix this issue, but I'm willing to try if you can confirm you had the same symptoms.