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 modify a Pokémon's stats to give them the Nature's growth properties; the Nature itself isn't changed. For instance, an Amped Toxtricity could have the stat bonuses associated with a Low Key Nature, even if it technically has a different Nature. This is relevant for breeding purposes.
Toxtricity (Bulbapedia)
Hope I helped!