The UK doesn't have fully nationalized healthcare.
* GP surgeries (doctors offices) are private businesses, and may provide NHS services or private services.
* The NHS contracts out for a growing number of services, ranging from operating whole departments (e.g. Virgin operates the out-of-hours walk in centre for non-emergency care next door to the A&E/ER at my local hospital) to cleaning.
* The NHS rents out facilities to private providers, and individual NHS trusts may derive income from carrying out private work (though the proportion of this income is limited, and the revenue goes back into their budgets). This is in addition to NHS staff often renting NHS facilities to carry out private surgeries outside their normal rotation.
* Dentists are mostly private and generally dentists that do NHS work also do private work, and often mix and match with the same patient (e.g. the NHS does not reimburse more expensive types of fillings, so your dentist may recommend e.g. things like porcelain veneers on visible teeth and stick to NHS options for less visible work).
* Pharmacies are mostly private - the NHS mostly operates pharmacies only in hospitals etc. but they too can be privately operated.
* 10% of the population have private health insurance. This is mostly "top up" insurance offering additional services or faster turnaround for non-essential surgeries etc. than the NHS.
* Many types of healthcare services are mostly private. E.g. while you can get some types of cosmetic surgery on the NHS
As it stands, the NHS is the most cost-effective hybrid or private system in any developed country. Too cost-effective some would say; to the point where it comes at the cost of making things more miserable for patients. But that's what you get when we pay far less than any other countries at a comparable income level.