Body piercer here. First things first, a nose piercing (or any other cartilage piercing, for that matter) should never be performed with a gun. They are designed for earlobes (and shouldn't really be used on those either, due to sterilisation issues, amongst other problems), not for considerably thicker cartilage - the piercing stud that your daughter was pierced with is blunt compared to a needle, and causes unnecessary trauma to the area.
Go into a different piercing studio - one that pierces ONLY with needles - and ask them to switch the jewellery out for something designed for fresh nose piercings, such as a titanium nostril screw. Don't let your daughter attempt to do this herself as the new jewellery MUST be sterile due to the 'freshness' of the piercing, and it will be much easier for a professional to insert the new jewellery with minimal irritation (poking around in a fresh piercing can cause further swelling and lead to infection).
Once the jewellery has been changed for something more suitable, leave it in place for a minimum of 8 weeks, preferably 12. Clean it twice a day with a saline solution, and DON'T fiddle with it or touch it with unwashed hands.
If the school have an issue with the piercing, which does raise the question of why it was done in the first place, the safest option is to removed the jewellery completely and let the hole close over - taking it out and putting it back in over and over again is a recipe for infection and potential scarring.
By the way, this isn't a telling off in any way (just in case I sounded a but preachy!), just advising from a professional perspective 