'Naughty' and 'cheeky' both sound very BE, and not necessarily very bad, possibly bad in a cute or funny kind of way. To AE speakers they will sound a bit old-fashioned and quaint, like Victorian children out of a book such as Mary Poppins.
'Obnoxious,' 'rude,' 'badly behaved,' 'brought up in a barn,' etc. are things I would say of children who are just unpleasant to be around. You could also say they're just total brats, though that word implies mainly younger children, say, under 12 or so.
'Impertinent' and 'impudent' are both fairly mild, similar to 'cheeky' BE, and implying mainly talking back. Neither is very common in ordinary conversation in the modern day, though both are still understood in writing.
'Insolent' is stronger and can imply quite strong disrespect to an authority figure, such as a teacher, commanding officer, etc.
'Brazen' to my ears might be used more for an act than for a person, since a particular action is more likely to seem utterly shameless. Of a person we might just say that they have a lot of nerve or a lot of gall, but it's still usually in connection with a particularly offensive act, not as much just as a description of their personality or character.