A pilot in this situation went through our ATO a few months ago, they completed training as required to pass the SEP proficiency check.
We recommended conversion to a new UK FCL licence. For this conversion, make sure you've demonstrated proficiency using radio aids and this has been certified in your logbook.
After you've passed the check, there's a fair bit of paperwork.
SRG1104 - Converts a previous UK issued licence to a UK FCL licence
SRG1119C - Renewal of an expired Class Rating
SRG2199 - Examiner Report
SRG1199 - English Language Proficiency (I think we may have used SRG5012 because it was easier to send with the rest of the paperwork, it wasn't rejected)
Then send all the paperwork, your logbook and certified copies of your ID/licence to the CAA.