Identify this bathroom extractor fan?
NuTone likely doesn't have anything to do with this.  That is a newer NuTone fan cover that has been drilled to fit over the center post on what is a very old vent fan.  That vent fan used to have a round cover that likely broke at some point.

That whole unit needs to be replaced.  The fan and housing look over 50 years old - there's just no point in trying to mess with what you can buy new for $15.