The paperwork has to be completed for both the FAA and the CAA and then submitted to both. The FAA then contact the CAA who respond and send back a confirmation that your details are correct.
When the FAA have this they will write to you and tell you to make an FSDO appointment. You do this, turn up at the appointed time and collect the temporary certificate. With this in hand you had for a school and do a BFR and that validates the certificate. A few months later the plastic certificate is issued.
Your class of medical or licence type does not matter, you will just get issued a PPL 61.75 certificate. It is up to you to keep the underlying licence valid as well as keeping the certificate valid with a BFR.