I too am having this problem. New iPhone 11 with OS 13.1.2 and new Apple Watch series 4. The watch updated to WatchOS 5.3.2 and then says it's current. Contacted Apple Support. I had already restarted the phone, the watch and then both he phone and watch. I checked the Usage and made sure there was no Update pending. On Friday Apple Support had me unpair and then pair. That didn't work. I ran out of time and took a case number.
Contacted Apple again on Saturday, 10/12. I was handed up to a Senior Adviser and then another before sending to another who called me on the phone. She concluded that we had tried all the trouble shooting they had available. She then had me take 3 screenshots (iPhone About screen, Watch About Screen, the software update screen showing the watch declaring software was current at 5.3.2) and upload them to Apple. She also confirmed my phone was at OS13.1.2. She got a confirmation from Engineering that they received the information and were working my ticket and the problem. She scheduled a follow up phone update for Tuesday, 7:30 am (PDT).
I will update any progress when it occurs.