Private licenses are indeed issued by a local office (Luftamt). You will probably also need a reliability check (ZÜP) to make sure you do not have a criminal record, drink driving convictions etc. This can take six weeks plus . So presuming you fly on your UK PPL till the end of the year I would start the process soon if you want to continue flying in the new year. You do need to allow for the fact that you will be a trailblazer and the local office is unlikely to have had much experience in converting UK PPLs.
It is obviously sensible to make sure you can get a class one medical before doing any hour building or further training. However whether a UK medical is acceptable to the Luftamt after the end of this year is by no means certain. I remember decades ago having a surreal discussion with my local Luftamt that my class 1 (German) medical met the requirement for a class 2 to issue a gliding PPL. Obviously it did, but it took a call from the doctor to persuade them. I tell this story not to make fun of anyone but to illustrate that there is a deep seated bureaucratic thoroughness when dealing with stuff outside their comfort zone.