Your ICAO papers, whether issued by the UK, the USA, or Mongolia, remain as before, and remain required by the State of Registry to fly an aircraft on that registry, worldwide. Nobody can take those away from you (except the issuer).
What EASA (an agency of the EU) has so far done is to force EU pilots to obtain
additional papers.
So pilots flying on
non-EASA ICAO papers, e.g. UK non-JAA PPLs, or FAA PPLs, or FAA IRs, etc, will need to obtain EASA papers as well - after April 2014 in the UK.
EASA could have brought in a reg requiring the pilots to wear pink underpants, or get a certificate in taxidermy. It would amount to the same thing.
For the portion of a flight which takes place in EU airspace, they have the power to do all of these things - because the UK signed up to obey the EU.
That is why I
did the JAA IR recently. It's not that I need it to fly, and in fact the UK/JAA PPL/IR is not even valid to fly my own plane (actually it is valid in the issuer's airspace - the UK - which is a bit useless, since the FAA CPL/IR covers me for that, and the rest of the world) but I got these extra papers as an insurance policy against what might happen after April 2014, or even before that should the 15hr IR conversion route disappear.