Eccleston grew up working class in Manchester, and he (understandably) didn’t think using his natural accent would be a problem. He reportedly butted heads with then-showrunner Russell T. Davies about his Northern accent and eventually left the show.
“I had a sense acting wasn’t for me because I’m not educated. […] British society has always been based on inequality, particularly culturally. I’ve lived with it, but it’s much more pronounced now, and it would be difficult for someone like me to come through. You can’t blame Eddie Redmayne, Benedict Cumberbatch and others taking their opportunities but it will lead to a milky, anodyne culture. To an extent, that’s already happened.”