You can work in the private sector as a junior doctor covering private wards and whatnot. However, you would just be doing the lackey work (and for any actually sick patients you would do what everyone else does - call an NHS ambulance) and your career progression just halts. You certainly wouldn't be seeing any neurosurgery, let alone doing it!
The only way to be a private neurosurgeon is to train to be a neurosurgeon on the NHS, then set up a private practice alongside your NHS work which as above, takes decades.