UK Travel rules as of 09/09/2021. As always, get up-to-date information directly from UK Government website [here] and be sure to check the arrival restrictions/rules of the country you are entering before travelling.
You must do these things whether you are fully vaccinated or not.
If you are fully vaccinated
If you are not fully vaccinated
If you do not qualify under the fully vaccinated rules, on arrival in England you must:
Test to Release
If you need to quarantine, you may be able to end quarantine early if you pay for a private COVID-19 test through the Test to Release scheme.