You need to identify which State you wish to be the State Of Licence Issue (SOLI) and start the process from that end. They then ask the UK CAA for your records. It doesn't apply in your case but others may wish to note you have to have an EASA licence to do this, if you had a UK only licence then step 1 would be to get the UK CAA to magic it into an EASA licence.
I don't know how you could get a second EASA licence from another State. My understanding is that all EASA licence records are held at the SOLI. I don't see how you could have, for instance, two EASA ATPLs. Looks to me that the last phrase of the CAA advice may not be fully thought through, but who knows, its all new territory?