There is a central organisation that sets overall directions and can give orders. But then there are lots of other organisations that are within it or attached in some way or even privately owned groups that operate in association with it.
Each group usually creates its own web sites and the central organisation doesn't usually interfere unless something is wrong.
The following is AFAIK correct, but it's my personal experience, and I'm not claiming to be an expert on the NHS!
As a simplified example, I am covered by the NHS. In my case NHS Scotland, which is separate from the similar system in England or others in Wales and Northern Ireland.
I am registered with the local NHS GP surgery which puts me on the system and provides most non-emergency access to care.
That GP's surgery is (probably) a private partnership of doctors who own the building and employ themselves and the other staff (nurses, admin, etc).
They get paid by the NHS system as a whole to provide me healthcare, but are not direct government employees.
I actually picked this GP's practice over several others, each of which have their own websites, because its website showed that it was open to new patients and because they offered open clinic times that matched my requirements.
On the other hand, the local hospital is owned (2) by the regional NHS hospital group as a government owned facility which employs the staff there as direct NHS/government employees. Except in emergency, I would go to my GP first, then be referred to the hospital.
However this is a separate organisation from the regional hospital group that operates hospitals in the next major city.
(1) NHS structure and funding: https://www.kingsfund.org.uk/sites/default/files/2017-10/NHS...
(2) Though see https://en.wikipedia.org/wiki/Private_finance_initiative for a more complicated explanation