I've traveled 2 times now by air from NYC -> London during Covid.
Last year, a 14 day quarantine on both sides, and on the NYC side, a phone call everyday to make sure I was quarantining.
This year on the London side, a 10 day quarantine, 1 negative test before flying, and 2 at home tests while quarantining, with a phone call everyday to confirm. I also got hassled at the airport because I didn't print out the day 2 and 8 receipt, just included the code on my passenger locator form under penalty of perjury.
So, the UK by air does care I suspect because of the increased exposure risk.
I traveled in the opposite direction (more or less: Vienna to Bucarest) more than 3 months ago and wasn't asked either, but on the Hungary-Romania border, they put me in a tiny room (1/3 shipping container or so) with ca. 10 people and gave me quarantine papers. They didn't check bags.
I'm pretty sure they'd check for a test or vaccination now on the way back, I'm not eager to find out and will stay here in the Covid exile for a little longer.
I know people who travel regularly and no one says anything to them. No quarantine, nothing. Most people on the streets here in Germany give zero fucks, groups of 5-10, no masks (even though there's a curfew between 10pm-5am due to an increase in cases).
And I traveled all the way from Moldova to Germany by car. No one, and I mean no one ever asked for a Covid test (which I had, of course, along with a positive antibodies test). Border guards, Hungarian border patrol, German Zoll, regular Romanian and German police, they cared about what's in the bags, didn't even mention Covid.
Hungarian border guards wanted to see a German work contract for whatever asinine reason, but I repeat, none of them ever asked for a Covid test or to at least keep the mask on (they need to see your face when checking the passport).
Whatever bureaucratic stumbling blocks there are, they have nothing to do with reality, it seems.