General practitioners and dentists usually operate as partnerships(*) who accept NHS patients (for GPs who subcontract to the NHS they pretty much have to accept onto their books anyone in their designated catchment area). In those specific instances a tag such as NHS_patients=yes would be appropriate.
However the situation is different for hospitals. The majority of them are funded and run by the NHS. A few might have a private wing or offer private treatments. For these NHS is the appropriate tag. There are some private hospitals around and NHS would be inappropriate for them unless they are co-sited within the grounds of a larger NHS facility but offer a limited range of service — typically quick and dirty operations. Even with the presence of a private hospital the site remains NHS.
(*) There are some GP practices that have been taken over by avaricious companies — predominately subsiduaries of large US health care comapnies — but they still accept NHS patients.