I have triple citizenship and currently living in London but my mother still lives in the US and I went there this past summer as she was alone during all the lockdowns. (She is a cancer survivor so once we were both vaccinated was when I traveled to see her.)
For the test to the US (which I took in June), I used Express Test in the UK. They were really professional and I had my results the following day at 10 PM. I would recommend them: https://www.expresstest.co.uk/
It doesn't matter where you do the Day 2 test, and there are drop boxes pretty much everywhere on the websites. I did mine with Randox Health when coming back to the UK next month. There was no issue coming back.
I am planning on going to Italy in two weeks and I will likely use their services again.
I flew with Delta (the airline, not the variant) back to the USA and they accepted my Express Test without an issue.
As a note whilst flying into the UK, as I can say I've done it twice during the pandemic, don't read too much into the chaos at Heathrow Arrivals. I might have been lucky, but I am also organised. I had no issues coming back into the country. There were no lines/queues and no one to check my documents. They did check twice in the US for my departure, once at the check-in desk and again at the gate desk prior to the international flight. I did see some people removed, but as much as I did try to eavesdrop, I didn't understand fully why they were removed. One of them was an American (I think American citizenship only) and Virgin Atlantic staff was telling him that he only had one of two of his vaccinations and that the one he got wasn't approved, so again I wasn't privy to reading his documents.
My suggestion for anyone travelling in these times is to have printed copies of your documents (COVID entry forms, vaccination status, test results) just to be safe and they are handy to have. I saw a woman at the gate desk whose phone's battery was dead yet she had all her docs on there.