I don't think 14 is necessarily too young, I know some 14 yos who act like they're still 9yo, and some who are going on 24 (maturity wise). How'd she'd react to requests to remove it, like school or future employer (can be h&s issue) would be a good indication.
That said, if the request was a bit out of the blue, I'd probably say something like, as it's a mostly permanent body modification, if you still want it at Xmas/next summer, we'll get it done then.
And I'd expect her to have researched aftercare, things that can go wrong, and found a reputable piercer (that I agree with).
Is your mum vaguely religious? IIRC nose piercings are mentioned in the bible as a symbol of wealth and status
I have multiple piercings, I think I was up to 5 before I turned 15 - even after I'd convinced them not to say "no", my parents made me wait 18 months for anything other than the basic ear piercings, but after that they fully supported me and joked that I was easy regarding birthdays/Xmas, as I either wanted another piercing, or later on, a tattoo
I hate septum piercings personally though, I know about their history, but as someone who has allergies all year round, it just makes me shudder