Which medical school you go to ostensibly has no direct bearing on specialty applications, particularly given that specialty recruiters are blinded to your medical school to ensure no bias.
Also you should be aware there are fewer neurosurgical consult posts than upcoming trainees and so neurosurgeons who have completed training now end up having to work in trust grade posts or do fellowships until a consultant somewhere ends up retiring, and then they all apply for that post. This is due to an error in workforce planning about 10-15 years ago where it was expected neurosurgeons would take on some then new procedures (I think to do with coiling brain embolisms or something?) but these ended up being taken on by interventional radiologists instead, so too many trainees were brought in to cope with an expected increase in number of cases, but as those never did increase, there was never a corresponding increase in consultant posts.
Basically you may end up earning much less after qualifying as a neurosurgeon than other surgical consultants because you probably won't end up in a consultant post. Neurosurgery is also generally one of the least "translatable" specialties to working abroad as most countries have enough of their own trained neurosurgeons and have no need to recruit neurosurgeons from other countries.
Realistically, as you need to study the whole of medicine for 5-6 years, then work across the whole of medicine for a further 2 years minimum, you probably should not be deciding on a very niche specialty already as you need to be motivated to learn about and work in all areas of medicine for the next 7-8 years minimum.