The CAA reference that you quote says:
Information for previous UK licence holders and holders of EASA licences issued before 31 December 2020
However when you read the page there is no mention of any process for a non UK EASA licence holder to convert.
The conversion process is however contained in
Regulation 2020/723 Careful reading of this document reveals only details of a Validation for one year or Conversion of any licence to a PPL; the transition to a Commercial licence is unclear and conversion details in CAP804 Section II Part P seem outdated.
Only last week the CAA notified an ATO that the conversion process was now purely administrative to convert an existing 3rd Country ICAO licence to a UK PART-FCL Licence. The person on the phone failed to give his name or provide any reference to this so should be treated with a deree of caution atributed to all hearsay.
With regard to the ability to use an EASA licence with a certificate of validation downloadable from the CAA website for the next 2 years is subjuct to the additional proviso that the 2 year validity is only applicable up to the next revalidation of any rating in the licence and possibly the medical certificate.