The "no accent" thing with English people means an accent a not-English actor puts on if they are playing an English part - "Estuary English".
I spent a huge chuck of my childhood in England and have the regional accent from there. It's one that everyone (in England at least) seems to hate too!!! It's not strong though as I've been back in Ireland a long time now. People here think I've an English accent alright but not everyone can place it to the right region.
I think for most people their accent gets "imbedded" in childhood. (But not always, years ago I met a woman in Birmingham who had a very strong Birmingham accent - told me she was from Crumlin moved to Birmingham in her 20s.)