But doesn't that mean the UK and the US operate in parallel on this?

If you fly EDI-LHR-JFK, then you're considered to have cleared security to the level required by the CAA at EDI, so no need for a rescreen at LHR, but in the reverse you're only cleared to FAA standards on arrival in LHR, so need to be re-checked.

The SYY example is a good one, but I have similarly flown GTF-MSP-DTW-LGW and only cleared security at GTF. Am sure there's other even smaller airports out there doing the same...