Jokes aside; it depends on where in the country you are, and what the current circumstances are in your own health care authority.
I’m a GP in a small country surgery in south east England. Our patients have great (open) access to us and our local hospital trust (where my husband is a consultant) is creaking but the emergency/ cancer/ urgent cover is really good. Not necessarily comfortable (long A&E waits) but safe.
Private health insurance in these conditions wouldn’t negate the areas that are struggling (emergency care) and wouldn’t improve your emergency access.
For lesser life threatening things; sore knee affecting function, funny neurology, need (or indeed want) an MRI for back pain, need physio, skin condition, gastro symptoms with no red flags (after primary care investigation) etc you might wait 8months + for a specialist NHS appt. in these situations insurance is super helpful.
My dad was diagnosed with a very serious cancer in NI last year (an area reported to be really struggling with healthcare). The first 6 weeks ‘waiting’ for scans and tests were awful but really wouldn’t have been much faster in private sector… maybe a couple of weeks quicker. My parents understandably looked into and indeed were desperate to use their expensive BUPA to get on with things; then suddenly all the results came in and the NHS swooped in, his care was second to none medically speaking. No bells and whistles but really just the same drugs and scans and surgery he would have had in the private sector.
In our house we don’t have health insurance. I trust the system in life risking or urgent presentations. However we have paid privately for the odd specialist consultation re the kids (an allergy appt; £400 incl patch testing, a specialist appt for 10yo still nightly bed wetting)… but always paid just to expedite appt, not because the care would not eventually have happened. Indeed both appts were with the same specialists who also deliver the local NHS service (working full time NHS hours) so really we were paying for a timely weekend appt rather than better care.
if your family are young, in good health etc personally I would favour setting aside the £100 or so aside into a savings account and accepting that if the NHS is falling short of your needs you dip into that.
Having said all that I know the NHS is struggling at different levels across the country so it depends on what your local service is doing.
Sorry- long post!