Whilst not all practices are able to see students on the NHS, many practices will happily see you on a private basis. The important thing here is to find out if the charges are reasonable and affordable in advance of your visit. Find out how much a check-up (including x-rays) will be and make sure they are then able to give you a written estimate of work that needs to be carried out. Shop around, check practices' websites and ask friends if they were happy with any dentist they might have seen.
Private practices are more likely to offer white fillings than metal amalgam fillings and will be more willing to help with cosmetic procedures such as tooth whiteneing.
Some practices have a dedicated scheme with reduced prices for students such as at the Redland Road Practice. Click on 'Further Help' on the top tool bar for more details of the Redland Road Practice.