The lack of EASA reciprocity is nothing to do with the licence you hold, or the CAA. This is bound up in political dogma in the EU.
I think you may find it was the UK that voted to become a third country. As everyone apparently knew what they were voting for, then it follows that they knew what being a third country meant. The degree of hardness of the brexit chosen (and for the purposes of this thread, the decision to leave EASA), was a deliberate choice of the party elected into parliament with an 80 seat majority.
So I would say that it is far more to do with the political dogma of the governing party in Westminster than it is the political dogma of the EU