I don’t know of any UK private hospital offering a true emergency service, so as far as your immediate fracture care goes you have no choice other than the NHS. However, subsequent care such as outpatient appointments with orthopaedic surgeons and any non-immediate surgery that may be required can often be taken care of privately.
From my own experience (as Consultant Anaesthetist who works in both the NHS and private sectors) going private certainly means you get appointments in a very timely manner and at your convenience, and the hotel side of the hospital experience should certainly be a much more pleasant. But as far as the actual medical care goes there really shouldn’t be much difference between private and the NHS. In fact if you are seriously in a bad way the NHS is the only place I’d want to be.
As for which private medical insurer to go with, I suspect they’re all much of a muchness from a patient’s point of view (although not from my point of view!). You could try some of the lesser known ones such as Exeter Friendly, WPA or Cigna.