If I ask for blood tests for children that can't be done by our practice nurse, the hospital phlebotomy department will usually arrange to do them within a couple of weeks.
If you want your dc to see a paediatrician privately, then just ring your GPs secretary and ask them to refer you privately, you will be expected to choose which hospital you wish to go to. They may be able to refer you to the same paed if they do work at the local Golden Nugget or may be able to recommend someone (GPs aren't supposed to recommend for private referrals directly). If you know anyone else who has seen a local paed either privately or NHS, then a word of mouth recommendation is often best.
As an aside, if you just have the bloods done privately, then you may find that if you see an NHS paed that they will be repeated again as different labs can sometimes use different reference levels for results.