Yes, you absolutely can. If you cannot get an appointment quickly enough or are considering a treatment that is not available on the NHS, such as tooth whitening, you can choose to go privately. You can then return to your NHS dentist once you have finished the private course of treatment. You will be registered at both practices and can access dental care at both.
However, you cannot receive treatment for the same issue from multiple dental practices at the same time. This means that if you have started a root canal treatment at a private practice, you will generally need to finish the treatment privately. If the issue causes problems again in the future, you can choose whether to have treatment on the NHS or privately.