I'm not in UK I'm in Australia, but I would imagine some things are similar with our health care system although each State here is a little different too and then within each State you have different Districts as well, making waiting lists vary greatly in the public health sector.

Private health care requires belonging to/paying for private health insurance and they usually have pre-existing illness clauses anyway that basically prevents claiming certain treatments if the illness or condition existed prior to joining the private health care provider. It is very expensive to receive private health care without having insurance.

Again, it may be different in UK however I imagine it's similar, but here there is also a priority or tier system regarding treatments and emergencies and that affects waiting times. It can take months and even years to get some treatments. Even the assessment for going on the waiting list can take a month and that's just an assessment to work out priority.