I have recently gone through the process. It was a minefield of complications. The hardest part of the whole process was dealing with all the bureaucracy. Harder than the exams.
I converted an ICAO ( South African) ATPL to an EASA ATPL. So a bit different.
1) Get a class 1 medical, I waited till the end of my exams, as did a bunch of mates and this is more time consuming than you imagine.
2) Register with the CAA and get your license verified and register with a ground school ( if you meet the requirements for an EASA ATPL you can be exempt from registration but you still have to write the 14 exams)
3) Attend some ground school and write exams.
4) Your practical training will depend on how much experience you have. Over a 1000 hrs may include as little as 5 hours. You will need to pass an English proficiency test ( what they don’t tell you and I found this out by accident , is that it can be done as part of your skills test.
5) mountain of paperwork ( more than you can possibly imagine) and wait and hope the CAA don’t mess it up. They cost me quite a few months of delays by messing up my paperwork.