I just entered into UK from EU, and went through the process and thought I would share what I learned.
LOCATOR FORM
Some time before you leave, you need to file a "Locator" form on the UK.gov site. This will be checked at the Eurostar line or at the airport check in desk before you will even be allowed to board.
You need your passport, where you will be staying, the code number for the 2nd day Covid test you will take in the UK, etc. This whole process is easier to do on a computer rather than a phone.
When you are done, you submit it. At that point, you can download the PDF of the document to your phone or computer. On Apple devices you can save these documents to the Notes app, which allows you to find it and show it even without internet.
If you need to make changes, you have to start over and do the whole thing again, and you will get a new reference number and new file.
COVID TESTS
We did our test in a little white tent set up outside Paris Gare du Nord. It costs 25 eu except on Sunday when it went up to 30 eu. It takes 15 minutes and they email and text you a result. (Although I never got the texts). The email has a link where you enter your birthdate. Then they email or text you a temporary password, to get access to your test result. You can add this result to your French health app wallet, or download the file, and then upload to Notes like the other document.
We had a family member in UK get at-home tests from Amazon. They give you a code number for each test. That code number is what you put on your Locator form.
VACCINATION STATUS
You are asked about your vaccination status on the locator form. And thats it. You just self-report. No code number or QR code or proof. I have a CA QR code, and both the French and EU health passes. But it was all on the honor system.
ARRIVING IN UK
I expected a strict interrogation on arrival. Years ago, I was questioned pretty rigorously when arriving in UK for 2 weeks to visit some universities which surprised me.
But this time, it was too quick and easy. We arrived and got in a line for US/Canadian/Australians. We scanned our passports in machines which seem to check that your face matches your passport photo. And that was it. More walking and suddenly we were out and free. No one checked any of our documentation - covid test, vaccination, locator form, etc. No one asked if we were bringing in drugs, exotic animals, Luis Vuitton knock-offs, etc.
We arrived about 9 pm at Heathrow and it was fairly empty and few employees - quite different than the huge lines I always experienced coming from the US to UK - often arriving morning or midday.
BEING IN UK
After months in the EU, we felt some peace of mind eating in restaurants, staying in hotel, or taking high speed train knowing that all of our fellow guests were checked for their vaccination status. Also, in riding dozens of metro trains in Paris, we never saw anyone NOT wearing a mask - except our last day when a guy was not wearing a mask, and he was being yelled at my another passenger for being a pig.
Now in the UK, no one checks vaccination in hotel or restaurants which is compounded by 20% of guests not wearing masks at all. Indeed, at our fairly expensive hotel in UK, even the manager walks around in the lobby talking to guests without a mask, and we had two maids in our room chatting away without masks. In most of EU, the employees and maids wear masks.
We find it unnerving to see this - while omicron is exploding. We risk not being able to get on our plane back to the US, and possibly a serious medical emergency in a foreign country. So we don't take this lightly.