You will have to present yourself to a HIMS(Human Intervention Motivational Study) AME (Aviation Medical Examiner)and he/she will defer your medical to the FAA in OKC. The examiner cannot issue you the medical as an initial issue.
Find a Senior FAA HIMS AME(in the UK if possible) who knows how to deal with the FAA on your particular issue. You will have to register with MedXpress and then schedule your appointment. It will help if you have all the required tests and reports that the FAA may require per the protocol which they have online. A special issuance medical can take time to obtain. Assuming you get everything right with the paperwork the first time around, it can still take a long time with some expensive testing/reports required. I would say 3-12 months is fairly common.
This might be of some help:
https://www.faa.gov/pilots/amelocato...int_part_2.pdf
https://www.aopa.org/news-and-media/...and-depression
https://www.faa.gov/about/office_org...tification.pdf