It's really up to you how you want to register. But if you register as an NHS patient you'll be given NHS treatment. This will mean that certain things might take a bit longer (the dentist might use a different lab for NHS work v. private work) and certain things like colour matching if you have a bridge, etc. will be slightly different. An NHS dentist cannot refer you for private treatment (to him/herself or to another dentist) unless what you need isn't available on the NHS (which is very rare).
Many NHS dentists also do private work, but may not be accepting new NHS patients because in their contract with the Primary Care Trust they will have a certain amount of "dental activity" that they can perform each year - once they are near their limit they might close their list temporarily.
The Primary Care Trust in your area has an obligation to try to find you an NHS dentist if you want one, though you might have to wait 6-8 weeks for an initial appointment or go out of your area if you're in a hurry.
NHS Costs are:
£15.90 - examination, diagnosis, x-rays, advice on how to prevent future problems, scale and polish if needed.
£43.60 - all of the above plus fillings, root canals, extractions
£194.00 - all of the above plus crowns, dentures, bridges