The issue here is ever since one of the iOS/WatchOS updates recently (can’t remember which one exactly but I think it was iOS 15/WatchOS 8), the AirPods Pro stopped auto dual connecting to the iPhone and Watch. Apple has yet to address this bug. Used to be able to just put the AirPods Pro in while wearing your Watch (while connected to your Phone) and any audio from the iPhone or Watch would route to the AirPods Pro. Now you have to manually connect the Watch. Swipe up to open the Control Center on the Watch, tap the AirPlay button and connect your AirPods Pro. Then they will be connected to both iPhone and Watch.
An example of this is if you have your iPhone, Watch and AirPods Pro on, and you get a phone call, if you answer on the Watch, audio will come only through the Watch unless you now manually connect the AirPods to the Watch. Before they’d automatically connect and answering on the Watch would route to the AirPods.
If you don’t believe me, try it out. Have your Watch connected to your iPhone and put your AirPods in. You’ll see them connected in the Bluetooth settings on the phone but if you go to the AirPlay screen on the Watch (from Control Center), they won’t be connected. Connect them on the Watch with AirPlay in the Control Center and you’ll see they will now be dual connected to the iPhone and Watch.
Can’t believe this hasn’t garnered more attention and fixed by Apple by now.