Helth Cover

Are there separate hospitals for NHS/private patients?
If you had private healthcare and were in a serious accident where you needed to be rushed to hospital, would you be taken to the NHS hospital and then transferred or taken straight to your private hospital? What if the injuries were so severe that they needed to be treated/operated on straight away, would that just be done in an NHS hospital?
If, say, someone needed a leg amputated, would that be done privately or on the NHS (assuming the person had private health insurance).
Would appreciate any information, thank you.