I'm not sure what the O.P should have done but to throw my hat in the ring.....
A few years ago my wife had a major workplace accident, horse kicked her in the head.
As part of her accreditation as a professional coach,the equestrian federation have their own insurance.
We also have top private health.
She was taken to a&e. Scanned etc. Major skull breaks and fractures etc etc.
To cut a long story short, as i remember, initally private health covered the airlift, nearly 40 hours of surgury, top surgeons, plastic surgeons, anethitists,etc etc etc and then they(the private insurer) chased the insurer from the equestrian association (AON). Not all was covered by AON and but the rest was coved by private.
While it was terrible at the time to have deal with every thing, not having to wait in line in the public heath system was gold. If it had to be done, it was. Both insurers were fantastic. They comunicated with each other and we wern't hassled unless nessasary.
To be able to just say private heath and give us the best treatment there is, was gold.