Your medical records are always held by the State that issues your licence. When you have a JAA Medical the AME sends a copy of the medical to that State. Under EASA it will be exactly the same, you can still get a medical with any EASA AME regardless of who issued your licence, they will simply get the report from the AME.
Is it possible to simply visit a UK medical examiner, take a class 2 renewal, and walk out with a new UK medical?
Yes or any other JAA AME.