Cost. There’s an NHS A&E within easy reach almost everywhere, and why would anyone want to pay more money for what they already paid for from taxes?
How could a private unit compete with NHS ambulances which aim (and usually succeed) to be on the scene in 8 minutes, which have motorbike paramedics who can get there even sooner, which have helicopter-born paramedics and off-road response teams some of which even have jetpacks for use in hilly areas?
Also, A&Es can only work with appropriate linked emergency services in the same building – e.g. CCU, ICU, dialysis and specialist surgery – all available 24h a day. Private hospitals simply cannot afford that, or need it often – especially as they look for ‘low risk patients’ that wont need all of that.
To finance such a thing a private hospital would have to charge US-level prices. And who on Earth would willingly pay such a ridiculous amount when they don’t have to?
Private hospitals in the UK are largely for elective care which the NHS doesn’t cover (or offer in good time) – e.g. for getting seen faster for low-risk ‘cold’ surgery (e.g. hip replacements and cataracts); fertility treatments; for psychiatric care, for which the NHS is poor and sadly over-stretched; wholly vanity-based cosmetic surgery; and for people who just want a more luxurious room and menu.
So most UK private hospitals don’t have any emergency care, unless they are the sort of super-expensive place patronised by Russian oligarchs and Saudi princes. Even then, if they have had heart-attack whilst in a private hospital, they will be rushed to the nearest major NHS hospital with an A&E.