You should get your teeth checked by a dentist every six months, and so if you are staying in the UK for a long time, you may need to register with a UK dentist. In Britain, there are many dentists who offer NHS treatment and equally, many who are private.
NHS treatment is cheaper than private treatment, but is not free. If you are eligible for full NHS treatment, you should register with your doctor first, otherwise you will have to pay the full cost. You should give the dentist your NHS medical number from your medical card. Remember that if you are eligible for free prescriptions, you are also eligible for free dental treatment. Similarly, you may be able to get a refund on the basis of low income. You can find your nearest NHS dentist here.
If there are no NHS dentist near you or else you would prefer to see a private dentist, then you can book an appointment with them simply by ringing up your local surgery. However, they are considerably more expensive than NHS ones. This increase in price does of course provide you with a better service, as you are likely to get an appointment much faster.