In britain these are the government funded schools similar to the american 'public school'. state schools are free to attend as they are funded through taxation (even if your kids go to private school, i know, soo unfair).
State school pupils fall into many catagories:

1- the middle class kid who's parents don't love them enough to send them to private school and would rather buy an aga instead.

2- the common chav ( aka scum, thief, rapist, slut, and any other annoying scummy person). key identifying points are the classic disfigured chin due to inbreeding by their now incarcerated father(who is also their uncle)and as much fake merchandise that they can fit on their face and fingers.

3- the 'individual', these folks usually spend most of their time hanging around under a tree, in the music, art or drama department proclaiming their individuality whilst dressed the same and listening to the same music as their other individual friends.

State schools are not bad places really, they offer just the same education as a private school just with added sound effects of gang rape behind the bike sheds and teachers glugging down cocktails of meths and copper sulphate to pass the day quicker.

The Nussy, or the “nose pussy”, if you will, was discovered during the corona virus pandemic of 2020. People that had to be tested for Covid-19 had to have their nose swabbed right where the brain connects, which often led to people rolling back their eyes and gagging.

A nose-swab-fetish developed from this, because we, as humans, ruin everything.

State school may also be used as an adjective to point out a stupid mistake someone makes.