What do I need to do to ensure the longevity of my Welsh dresser?
No. Keep it in a house (i.e. a dry, somewhat climate controlled environment) and it will become an antique, or more antique, in the old fashioned way, by existing. 

Waxing/polishing are purely cosmetic, and should be done before your Mum visits to keep her happy. 

The basic construction of wooden furniture dictates its odds of becoming antique / more antique. Once it's gotten to antique status, short of wood eating pests it's generally good to become more antique until some external event ends it. So make sure your smoke detectors are working, etc...