Just flew back to SFO from Munich via Heathrow and can confirm the testing requirement is a bit vague for the airports abroad. Munich didn't know or care since we were connecting through Heathrow. LHR was a cluster with security and immigrations lines either duplicating requests to see vaccination or had no clue.
In the end, we got stickers at passport control and also a second check by the gate agent.
We did the remote Binax tests via video call. Pretty easy but you have to have a stable internet connection since they "watch" the testing process and 15 minute results period