The following is a BBC interview with the British Transport Secretary on foreign travel. It is about travel outside the UK for residents of Britian. But before Americans can travel to the UK, it's people will more than likely be allowed to travel out.
Too much information--and too complicated--to try to summarize here. But here goes....before Americans can travel to the UK without quarantine, we would have to qualify for their "green light" category. We have a ways to go. No word on any special rules for people who are vaccinated. Reviews of these rules will take place on June 28, Jul 31 and October 1 at the latest. This is just for England. However, this can still change at any time.
Remember, also, that the UK is no longer a part of the EU and EU rules don't apply.
BBC News - Covid-19: People can start thinking about foreign travel - Shapps