Is it possible to keep it registered in Greece and fly it in the UK
Microlights are entirely the responsibilkity of the National Authority(NA) of registration, and each NA can only authorise flight in its own country.
You can get the UK CAA to authorise it's flight in the UK on a Greek registration for a short visit, but I wouldn't hold my breath waiting for an authorisation for continuious operation, if I were you.
MJ
Ps. Brave New World?
Not quite the words I would use to describe the present state of European GA!