Assuming you're an existing NHS patient at your dental surgery then I'd suggest just asking them.
If you aren't registered as an NHS patient already you may find it hard to do so as a first place, especially if trying to register as a paediatric NHS patient. In many regions few if any dentists are taking on new NHS patients and even fewer seem to be taking on new paediatric NHS patients.
So it may well be whatever you're entitled to under the NHS dentistry scheme, you may not be able to access that and have to go private regardless if not already registered as an NHS patient.
NHS dentistry isn't like medical care - its not always available by default. In many places (e.g. where I live) there are no dentists taking on new NHS patients of any kind and the waiting lists are extremely long, necessitating people often needing to pay for private dental treatment (which is very expensive although with insurance or dental pay plans can be more manageable). If there is a dental school near you though you may be able to get free dental treatments for routine treatments by dental students -they often advertise such schemes if available.