I can only comment on my own personal experience and certainly I'm not qualified to give advice. I've never had a problem renting a vehicle in the UK using my UK licence (which bears a UK address). I've never been asked for any documentation as proof of address which, in any case, I couldn't provide as I own no property in the UK. I've only ever presented my licence (including the paper part when that was a requirement) and Passport for ID. I could have provided proof of address in Greece but have never been asked for that. I haven't tried to rent a vehicle in Greece since the paper part of a licence was discontinued and don't know the system for allowing the rental company to check for driving offence recorded on a licence.
As said, that's just my experience and not advice to risk attempting a rental without proof of address.