From the 4th October, the UK Government simplified the rules for entering the UK, and it is all based around whether you have the vaccine or not. This is providing you do not travel from or through a red list country. The red list countries can be found here.
There are still some hoops to jump through if you are fully vaccinated, but it is now more straightforward. If you have not had the vaccine, this does not stop you from entering the UK, it will simply increase the number of steps you have to follow before getting on the plane. Please ensure you read the guidance on the GOV.UK website before you travel, and remember to check the differing specific information for England, Scotland, Wales and Northern Ireland.