It doesn’t make any difference whether you have been vaccinated or not. Bermuda is not currently on the “red list” of countries which require you to quarantine in a hotel on arrival in the UK. However, you will still have to follow the general quarantining rules.
To be allowed entry into the UK, you must be able to show proof of a negative Covid-19 test taken within 72 hours before travelling. On arrival, you must isolate for 10 days at the UK address you provide on your passenger locator form. You must not leave your house for any reason, not even to buy food, medicines or other essentials, or for exercise. You cannot see friends or family from another household during this period.
You must also take a coronavirus test on days two and eight of quarantine, at a cost of £210. If you test positive, you must self-isolate for a further 10 days. There is a £1,000 penalty for not taking the test, followed by a £2,000 fine for failing to take the second one, with quarantine automatically extended to 14 days.
Under the “test to release” scheme, travellers from countries not on the red list can choose to take an extra test on day five of isolation. If you test negative you can stop isolating. If you test positive you must quarantine for 10 more days after the test. Anyone using the scheme still has to take a further test on the eighth day.