As far as I know, you only need a passport from the country where you are currently a citizen. If you're a citizen of both UK and Greece and have a passport from both, then you should only need to take one with you.
For instance, when travelling from the UK to the US, I only need a UK passport because I am a UK citizen. The passport's purpose is to identify who you are and where you're from.
I would probably advise against using one passport when leaving the UK and another when arriving in Greece, this might in fact flag up something nefarious going on. (just speculation on my part).
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