It will depend on your insurance and how it works in the UK. In some areas here are private GPs who accept insurance, and private hospitals that you can visit without accessing the NHS for a referral. We have a worldwide insurance policy (from the US employer) which allows us to self refer to specialists at local private hospitals in the UK, and we can visit either the private GP or the regular GP, whichever is most convenient.
If you have a UK policy with, for example, BUPA, I believe you do need to visit the NHS GP for everyday care but you can self refer to a specialist. Your GP will treat your child so you don't need to find a pediatrician. There may be a GP at the surgery/health centre who specialises in children but, in general, your GP will treat the whole family.
What ksand24 is referring to is if you have a non-urgent condition and don't want to wait for treatment. It is possible to visit the GP for a referral, see a specialist privately, and pay for private treatment, thus eliminating the NHS waiting list. This is not the same as having insurance.