As reported earlier, May 17 is the date England, and much of Scotland, eases lockdown rules. The U.S. wil be amber.
The G7 summit takes place from Jun 11-13 in Cornwall, England. Here the leaders of the US, UK, Canada, Germany, France, Italy, Japan and the EU will get together. Travel between these countries will probably be on the agenda.
On June 21, the British PM is expected to lift almost all lockdown rules in England.
I wouldn't be surprised, not thinking it is going to happen, but wouldn't be surprised, to hear about some deal for fully vaccinated Americans and British to be able to visit each other's countries without quarantine and multiple testing.
The word will probably come sometime between the two above events.
As for the EU, they are further behind in vaccinating so it will be interesting to see how they pull this off. But they probably will.
My guess, and it's just a guess, is that U.S. travelers will have to show proof of vaccination and a negative covid test BEFORE boarding the plane. Larger airports in the U.S are opening testing centers that offer quick results.