Will streaming service watchlists transfer when replacing old Roku streaming player w/ a new one?

Those channels you mention most likely maintain things like a watchlist in the cloud, not on the device itself. Roku devices in general have very little storage space, and channels don't store such information locally. If you can check your account on a different device (such as a web browser) and you still see the same lists, then you should be fine with replacing your Roku and still having all that data available.