In essence a fully vaccinated traveller is required to book a 'Day 2' PCR test (Lateral flow tests no longer allowed). This test must be taken on or before Day 2 (where Day zero is tha day of arrival). The traveller must quarantine at home or the place they will be staying until they receive a negative result.
Note the test can be taken anytime before the end of Day 2 - so it can be taken onj arrival at an airport testing facility.
Guidance for other parts of the UK may be different- please follow links from the 'England' page for these countries.