E.g. my old GP used to provide both private and NHS services (they were precluded by their NHS contracts from providing private services to people registered with them with the NHS).
Many NHS trusts also provide private services, as they are allowed to do so to improve utilisation and supplement their budgets, so in practice this is part of the reason the NHS is so cheap compared to universal systems in similarly rich countries.
Most private hospitals in the UK also e.g. rely on NHS for intensive care, and this, along with relying on the NHS for first-line care (A&E, GP's unless there's a wait, etc.) is also why private health insurance in the UK is unusually cheap, and why private hospitals in the UK are unusually cheap (if you're in the US, and planning elective treatments, it can be cheaper to fly to London and do it here, even factoring in hotels - and some Central London hospitals have hotel suites, and at least one have or had a previously Michelin starred chef because they cater - literally - to high-end international healthcare tourism).