If you're critically ill, and need urgent care, going private (with or without health insurance) doesn't secure you any priority.
Where it makes a difference is if you want elective care, or if you're chronically ill and require continuing care (and a high proportion of us will find ourselves in one or other situation at some time in our lives).
As regards elective care, while things vary depending on exactly what condition you want treated and on other factors, going private can often mean a shorter wait time.
As regards continuing care for a chronic condition, going private will generally secure you a more pleasant (or, I suppose, less unpleasant) experience.
Finding solid evidence of better health outcomes overall from private care is difficult. People who have private care do enjoy measurably better health outcomes but they would anyway, because as a class they are better off, and have a more advantaged background and experience, than the class of people who don't get private healthcare. The socioeconomic determinants of health work in your favour and it is difficult to disentangle the effects of that from the effects of receiving private care.