According to the GOV.UK website, fully-vaccinated individuals are permitted entry into England from outside the UK, but a Day 2 PCR test is required. This test may be taken any time within two days of entry into England - even immediately upon arrival at Heathrow. It also helpfully provides a filterable list of providers that will do Day 2 PCR testing, and inbound travellers are required to book such a test before departing for England.
However, there are options for a "self-test at your location" and a "self-test at a provider site", in addition to supervised tests. This surprises me. I am aware of two common methods of swabbing for PCR tests - the nasopharyngeal swab, which is inserted through one's nostril deep into the back of the nose, and the throat swab, which is inserted through one's mouth to the back of the throat. Unlike lateral flow tests (a.k.a. antigen rapid tests), both methods of PCR swabbing reach far deeper into the orifices of the patient's body, and often induce a bit of a gag reflex. This probably makes swabbing oneself difficult, dangerous or even impossible. In Singapore, for example, PCR self-tests are unheard of - you have to go to a clinic to get swabbed and the swab is performed by a medical professional.