To ask - what is probably - a very silly question

[ScribblerOnTheRoof]

Can I make an appointment with a neurologist at a hospital or must I be referred by a GP?

Referred.

Unless private, in which case you can just ring up and book, I imagine.

No - even going private you usually need a GP referral. You just tell the Gp you want to go private and they'll give you the referral letter.