PHE came into being on 1 April 2013 as a result of the Health and Social Care Act 2012. The Act did not specifically create PHE but gave the responsibility for certain health matters in England to the Secretary of State, who created PHE to discharge them.  Radiation is implicitly covered by section 11:

11 The Secretary of State’s duty as to protection of public health

After section 2 of the National Health Service Act 2006 insert—

 “Provision for protection or improvement of public health
2A Secretary of State’s duty as to protection of public health

(1) The Secretary of State must take such steps as the Secretary of State considers appropriate for the purpose of protecting the public in England from disease or other dangers to health. 
(2) The steps that may be taken under subsection (1) include— 
(a) the conduct of research or such other steps as the Secretary of State considers appropriate for advancing knowledge and understanding; 
(b) providing microbiological or other technical services (whether in laboratories or otherwise); 
(c) providing vaccination, immunisation or screening services;
(d) providing other services or facilities for the prevention, diagnosis or treatment of illness; 
(e) providing training; 
(f) providing information and advice; 
(g) making available the services of any person or any facilities."

This applies to England only.  Section 58 requires the "Appropriate Authority" in each of Wales, Scotland and Northern Ireland to take similar steps in relation to radiation:

58 Radiation protection functions

(1) The appropriate authority must take such steps as it considers appropriate for the purposes of protecting the public from radiation (whether ionising or not). 
(2) The steps that may be taken under subsection (1) include— 
(a)the conduct of research or such other steps as the appropriate authority considers appropriate for advancing knowledge and understanding; 
(b) providing technical services (whether in laboratories or otherwise); 
(c) providing services for the prevention, diagnosis or treatment of illness arising from exposure to radiation; 
(d) providing training; 
(e) providing information and advice; 
(f) making available the services of any person or any facilities.

In practice, PHE still seems to be exercising radiological protection functions across the whole of the UK despite the name.

PHE is an executive agency of the Department of Health.

Section 56 repeals the Health Protection Agency Act 2004 and abolishes the HPA.

HPA was established by the Health Protection Agency Act 2004 and came into being on 1 April 2005.

Section 3 of the Act gave HPA the following key responsibilities for radiation (the rest of section 3 deals with how they could be given extra responsibilities and who they had to liaise with):

3 Radiation protection functions

(1) The Agency has the following functions in relation to risks connected with radiation (whether ionising or not)— 
(a) the advancement of the acquisition of knowledge about protection from such risks; 
(b) the provision of information and advice in relation to the protection of the community (or any part of the community) from such risks.

Those responsibilities applied across the whole UK, though section 2 specifies how the Devolved Administrations could give them specific functions. The legal status was a "non-departmental public body".

The former NRPB became part of HPA, sometimes known as the Radiation Protection Division (RPD) but more normally as part of the Centre for Chemical, Radiation and Environmental Hazards (CRCE).

Schedule 4 of the Health Protection Act repealed the Radiological Protection Act 1970, thereby abolishing the NRPB.

The NRPB was created by the Radiological Protection Act 1970.  Section 1 gives the key responsibilities:

1.-(1) There shall be a public authority, to be called the National Radiological Protection Board (in this Act referred to as " the Board "), whose function it shall be- 
(a) by means of research and otherwise, to advance the acquisition of knowledge about the protection of mankind from radiation hazards ; and
(b) to provide information and advice to persons (including government departments) with responsibilities in the United Kingdom in relation to the protection from radiation hazards either of the community as a whole or of particular sections of the community.

The NRPB's functions were extended to include non-ionising radiation in 1974.  The NRPB functioned UK-wide. Its legal status was a "public authority".