Do you have to redraw polygons that have been uploaded to register them as having been surveyed?

No.

Uploaded polygons will be imported as pre-survey data. When doing a survey using the app you base your full survey on a pre-survey habitat feature. When you do this you copy the pre-survey geometry, you don’t use it directly.  This allows you to e.g. adjust the feature’s extent if required. Your pre-survey data remains untouched and another record (your full UKHab survey record) is created.