"Ferme ta bouche" (shut your mouth) or "ferme la" (shut it) are not mild, but not as strong as "ferme ta gueule" or "ta gueule". I don't know if there's a real difference between "ferme ta gueule" or "ta gueule", it probably depends more on how it's said than the words. Once you start using "gueule" it's a pretty strong form.

Really the two words "ta gueule" is shorthand for effectively saying "you have the mouth of an animal (a predator?) not a human, and you should shut it".

The closest English equivalent is "shut your hole".