I'd say either 'I was born and bred in London' or 'I'm London born and bred' (as a complete set phrase). If the person still lives in London now, they'd be more likely to say the second - in the sense of 'I'm a true Londoner.'
To be honest, I wouldn't worry about this phrase, and I certainly wouldn't see it as a grammatical mistake. Yes, the idea of 'I'm born..' does seem strange when viewed in isolation, but old-fashioned set phrases - like 'born and bred' - often do produce some odd constructions.