So Toxtricity’s form (Amped or Low Key) depends on its nature, when you use a mint to a nature required for the form, will the form change too?
No. Mints only edit a Pokemon's stats to give them the growth of the Nature; the Nature isn't changed. For example, an Amped Toxtricity could have the stat bonuses associated with a Low Key nature, even if it technically has a different Nature.
Toxtricity (Bulbapedia)
Hope I helped!