Flesh on the bones….
With my EASA TRE I can do LPC for all EASA licences providing I comply with the Examiner Differences Document. I can also do UK LPC providing I get pre auth from the UK CAA.
With my (hopefully soon to arrive) UK TRE I will only be able to do UK LPC, no notification required.
As rudestuff states, providing all the hoops are jumped through, one test, two signatures (also two lots of ppw for me!).
Hope this helps