I have just renewed my Blue Badge, Warwickshire County Council, renewal done online with the only document required was a copy of my PIP entitlement for the expiry date based on the PIP renewal date. The badge was sent from the central issuing authority niw. One word of warning, Warwickshire say it takes 12vweeks and sure enough it did take 12 weeks.
My council has always post 12 weeks too so I always put mine in 12 weeks early then at about 10 days before the expiry of the old one I send an e-mail asking if everything is in order and they always reply saying the same thing, “your application has been sent to the printers and will be with you soon”. Everytime, or so it seems, the badge turns up the next day.