It is very hard to find the law as it is dotted all over.
I found this Health Protection (Coronavirus, Restrictions) (Self-Isolation) (England) (Amendment) Regulations 2021
" Citation, commencement, extent and application
1.—(1) These Regulations may be cited as the Health Protection (Coronavirus, Restrictions) (Self-Isolation) (England) (Amendment) Regulations 2021.
(2) This regulation and regulation 2(2)(a)(i) come into force on 19th July 2021.
(3) The other provisions in these Regulations come into force on 16th August 2021.
...[these apply to England only...
These Regulations amend the Health Protection (Coronavirus, Restrictions) (Self-Isolation) (England) Regulations 2020 to ensure that a person can leave self-isolation to post an antibody test, and that the following people are not required to self-isolate if they have come into contact with a person who has tested positive for coronavirus—
(a)a person who has received a complete course of the coronavirus vaccination, provided that the contact takes place more than 14 days after they have completed their course of vaccinations,
(b)a person who has taken part in a vaccine trial,
(c)a person who can provide evidence that for clinical reasons they should not be vaccinated, and
(d)a child.
The Regulations also provide that, where a person comes within those categories, but was in a period of self-isolation on 16th August, their period of self-isolation will end at the beginning of that date, and make a number of consequential amendments.