Is it normal for the kneecap to be wiggly after total knee replacement? (1 week since surgery)

Potentially: I would discuss any concerns with your surgeon, but there certainly is some degree of healing that must occur in the patella after a knee replacement. The patella is actually inverted inside out to expose and replace its undersurface with a polyethylene bearing surface, then the medial aspect of the quadriceps mechanism, which helps in tracking of the patella, must be divided and then repaired.