It would be useful to know if the OP had a UK issued licence in the past. There must be some procedure in place, if you open SRG2158 the form can be used where:
The applicant has not previously held a UK issued Part-FCL/BFCL/SFCL Licence and/or examiner/instructor licensing certificate which was transferred to another EASA member state prior to the end of the EU-UK transition period on 31 December 2020, but they do hold a Part-FCL/BFCL/SFCL licence and/or examiner/instructor licensing certificate whose date of initial issue by an EU state is prior to 1st January 2021.
This would seem appropriate to the OP whose EASA licence was issued in 2013. The e-Licensing dropdown section says it is specifically not available to those who have never held an UK licence (which makes sense as it requires access to the CAA portal). If you follow the link for "Instructor/Examiner privilege conversion" you get to
this page. It states:
This is applicable for both ex-UK CAA pilot’s that have transferred out and EU pilots obtaining their first UK licence.
I agree, quite what that process is isn't clear, but assuming the applicant has instructor privileges on their EASA licence, the flow charts allows for transfer of those privileges to the UK FCL licence (subject to varying conditions).
I'd be equally cautious of any ICAO conversion being entirely administrative.
I've sent you a PM regarding the last point as I don't want to hijack this topic..