I am surprised that you, as the so called "expert" on here does not know about the limit on using private hospitals without prior agreement with the insurance company. I enclose a screenshot from the t&c of a well known UK insurance company. it clearly states its not for use for private treatment. Although I am able to find it, somewhere else in the document it states only government hospitals unless prior agreement is obtained.
I'm surprised that you didn't read your own link before commenting on her expertise. That extract does not state that treatment in private hospitals, as opposed to government hospitals, is not covered.