Reattaching the stud is future me's problem.
As I was scrolling down I thought "he hasn't mentioned repinning the stud yet!" and then there it was...
So the question I have is, how did the stud become separated from the balance spring? If it broke off, remember that when you repin it, it will now be shorter. Shorter = faster rate, so hopefully you have enough range to slow this down...
I did notice that generally you cant seem purchase the hairspring part separately - althrough ofrei offer a bunch of options. Am assuming that the balance wheel and hairspring are very closely aligned, and the usual (simpler) approach is to replace the whole part.
Likely I'll buy some hairsprings to practice adjustments on though, so do plan to come back to attempting a stud installation. - (unsure as/when i'll consider landing a hairspring vibrating tool to make that part usable)
Parts are typically sold as a balance complete which has already had the whole balance and hairspring assembly poised and vibrated. Not that its not possible to install a fresh hairspring, but as you have just learnt hairspring manipulation is a challenging task not to mention for most people pretty time consuming. We have only done the inner curve and outer curve manipulation on hairsprings at school so far, and its certainly been one of the more challenging assignments i've gotten so far, but i have also yet to do the dynamic poising thats a bit later.