Yes, you can still use NHS services whilst having private care, but you can't really flop between the two for the same condition. We looked at that with Niall's eyes. Thinking we'd get the assessments done privately because this was the hardest to get scheduled, then use the NHS for the operation (particularly as this would be the most expensive).
The Dr was happy to do that (he does both private and NHS), but once we switched to NHS for the operation, the remainder of his care would be carried out via the NHS. Of course your mileage may vary.
We still use the NHS for most things, but when the waiting lists for specialists are long, then we go private.