UK privately-sponsored healthcare specialises in things that you cannot get free from the National Health Service or things where the wait time is long - for example knee surgery. People don’t use private healthcare for emergencies or intensive care. For example the prime minister was in intensive care in April - in an NHS hospital.

Reading the full paper, the population surveyed was people registered for at least one year with General Practice doctors using SystmOne software. That covers 40% of the English population. It is probably safe to assume that almost everyone in England is registered with a GP. There is variance in the GP use of the specific SystmOne software - the paper mentions on page 13 that only 17% of London GP practices use it.

The quirk of the UK system is that private insurance companies have no legal or moral obligation to cover anything as you will always be able to fall back to the NHS. So getting cover than includes preexisting conditions or severe long term illness (e.g. MS) is very expensive and rarely negotiated by employers except in firms who can afford to do it.

Same in countries with public healthcare in the EU; when you have an emergency, you end up in public healthcare (which, in the countries I have lived in, is excellent), when you have a knee or hip operation, you benefit from private if you have it.

It would be surprising if in any EU country private clinics dealt with covid19 unless you're maybe super rich with personal doctors around you all the time.

Private healthcare providers in the UK don't seem to be providing hospital treatment for COVID-19 - this is left to the NHS, BUPA says:

"As a public health emergency, COVID-19 is being treated through the public health system."

https://www.bupa.co.uk/coronavirus/health-insurance

Not that private healthcare providers are actually providing resources to the NHS so its possible that you could end up in a private hospital even though you don't have private health insurance.

Currently private hospitals in England were requisitioned to cope with surge demand for NHS patients. They're going to reopen to private patients soonish.

https://www.bbc.co.uk/news/uk-england-london-52381675

This is pretty common in places with dual systems, my wife just had surgery on her shoulder in a private clinic but paid for by the public system.