Had this before (though as FD says, it's never a guarantee that it was correct).
Revalidation of your UK medical before the run out date would prevent your medical from "expiry", so the pedants who want to play on words would not be relevant to the argument.
It also brings up the reminder that, according to the UK CAA Flight Ops Inspectors (to whom I asked this very question, over lunch, last Wednesday) your UK licence is not valid if not supported by a current JAA medical - so you will need to revalidate/renew your JAA one anyway.
Seems like the pedantic way out of it - should you allow your JAA medical to expire before renewing it - would be to do both medicals (though what we had before was that the revalidated JAA medical was accepted as fulfilling the requirements).
Another one of those "play on words" arguments that could spend years in the courts if you have an incident.
I think englishal has the perfect answer to that one - get both from the same AME.