I am greek national and citizen living in uk under the eu laws; my son is born in uk not register in greece and never lived in greece; his birth certificate is register in uk; what nationality is he and is he greek or british citizen.

First of all, your son was born in UK, he obtains citizenship from the country where he was born, so he is British citizen.

Do you have dual citizenship? If yes, your son is entitle to claim Greek citizenship as well. You need to report birth overseas of Greek national at the embassy. You need to submit copy of birth certificate, marriage registration and photo ID of both parents.

After report birth at the embassy, your son can apply for Greek passport immediately.

He is both: he is British because he was born on British soil. He is Greek, because he was born to a Greek parent.

One note: he cannot apply for a Greek passport immediately after the birth is reported at the embassy. The process includes two more steps after the birth is registered since he is a male: 1) He needs to be registered in the military records [mitrwa arrenwn] in the municipality that you are registered in Greece, 2) After registration in the military records, he needs to be registered in the municipal logs [dimotologia] of the same municipality.

The Embassy in London can help you with making that happen.