I flew to the US on Saturday. I was never asked to show the UK form. I was asked at Check-in for a flight to Serbia a couple of weeks ago to simply show that I had filled out the form.
At first wing Check-in I showed my covid test (within 72 hours of departure time for flights to the US). Evidence of having Covid and recovered within the past 3 months is also suitable for the US.
At the gate I was asked to read and sign a form which basically said that I had a covid test.
On the flight I was given a paper form to fill out which might be reviewed upon arrival.
Upon arrival I had the luxury of both global entry and a BA Concorde team member. I was not stopped for anything.
In contrast with flying to the UK, where the covid test and declaration form were checked at Check-in, at the gate for each flight, and upon arrival at the UK border.
The US seems to be trusting that the airlines are doing the work, whereas the UK is verifying everything at every stage. Not that the UK verification was challenging, 30-60 seconds if you have everything in order.
Similarly if you are in transit landside in the UK, just choose the option on the form for airside transfer to not book the test kit, and then show proof of your flight the following day. Thankfully the border force is being pragmatic and not making you purchase a test kit when you won’t be in the UK that long.