.... Can anyone offer any advice or information regarding why you would be turned down and what life would be like without private insurance. ....
You'll wait a bit (or a lot) longer for treatment, especially if you don't have a life threatening condition, but if you need treatment you'll probably end up seeing the same consultant/ specialist whether you have private health insurance or not, just not in some fancy consulting room if you are relying on the NHS.
I doubt if many people in the UK have private health insurance
unless it is provided, or heavily subsidized, by their employer or trade union as a perk of their job.
Also see
here for other recent UK-Yankee discussion on this topic.