A few years back, I did conversion from FAA PPL to EASA with CAA. It took me 3 months to get the license although the papers were complete. As I fly on the continent, mostly in France, Germany and Spain and in view of the imminent Brexit, had to transfer the license to France. CAA promised to do fast in view of the special circumstances, it nevertheless took two months to transfer my medical records. It took the French DGAC just a couple of days to issue the license after receiving the records from the UK CAA. Besides, the French charge much less than the CAA for issuing license and the administration work.