I have a friend who currently is flying Biz jets, and he recently renewed his EASA license. It was reaching it's expiry limit which I believe is 7 years, if you don't renew within this time, you have to do all your ATPL's again I believe, but you should be well within these limits now.
When he did his LPC renewal at Flight Safety, he had a TRE, that was EASA approved, who then did the check for both EASA UK and FAA at the same time. I believe it was mainly a bit of additional paperwork to do this.
You need a TRE that is EASA approved, and have approval from your EASA CAA which would be Greece.
He conducted the flight test, and than he came over to the UK to do the EASA medical, it's possible you need to do a full medical as initial, as this is what he had to do, and then submitted all the paperwork to the CAA. Around 5 - 6 weeks later he got his license.
With your amount of hours you should be able to get an ATPL if you have done the skills test for this, or do this when you do your EASA LPC.
In theory you only need a flight check / LPC and medical. I am not sure if you have access to any EASA TRE's where you live and work. But that would be the easy way of doing it, on current type which you are already familiar with and just do LPC check with EASA approved TRE.