Bit of public information, but always check with the .gov website as the rules seem to change every week.
On of the problems in the past was that the UK would only recognise vaccines given in the EU and the USA and the UK of course, so even if you had been vaccinated outside of those place you still had to do the quarantine, pre and post arrival PCR tests. That's changed they are now recognising Astra Zenica, Pizer, Moderna given just about anywhere in the world now, make sure the certificate is in English.
Pre departure negative tests are not required, one of the few bits that made sense is not needed, no need to go into isolation on arrival another bit that made sense that has been dropped. Now you need to do a test on day two, still can't use the NHS testing system you have to book in advance a private test, this can be lateral flow. If negative you're free.
Not looked for a while but presume you will still need to do the on-line passenger locator thing.