Well, maybe I can help here.
My wife is in fact Robert Peel's Great great great great great great niece (if i read the tree correctly) and she does NOT have a Lancashire Accent. Nor does my Mother-in-Law who was born a Peel Cox. (True, but I'm not being serious, ok?)
John Peel, on the other hand, would have had a Cumbrian accent, and you might even get a chance to check that for yourselves if you dared. When filming 'Cumbrian Tales' for the BBC I was based in the fabled huntsman's actual house near Ireby - and my colleague, who lived there, had often heard the man clomping around (heavy steps in an empty house, on floorboards where there was now carpet). I never heard him myself, but odd things did happen...
Robert has never paid us a visit here in Leeds, which we think a little remiss of him. But if he does I'll let you know how he speaks.