Splitting hairs, but the public health bodies are separate from the National Health Services (so there exists Public Health England _and_ NHS England). Public health is slightly different to healthcare, but of course they work together.
Oddly, Public Health Scotland is part of NHS Scotland; but Public Health England is not part of NHS England, it's a parallel department.
Oddly, Public Health Scotland is part of NHS Scotland; but Public Health England is not part of NHS England, it's a parallel department.