If you want an answer from B'ham CC I would recommend you send a letter marked Private & Confidential to the Treasurer/Head of Finance (you should be able to find the correct title and name on their website). I would be sure you have all your dates in order, i.e. the date you sold the car and how soon the DVLA were notified. Normally a council will give you 14-28 days to pay and then they will contact the DVLA for the owners details. It costs the Council to find out these details so they will normally wait longer. I find it odd that they chased your sister at the right address even though you are saying they were writing to your old address? How did this change? If they asked the DVLA again, then surely they would also have seen the name had changed. A bailiff can only seize a car once they are satisfied the car does belong to the debtor which in this case it didnt, all your sister needed to do was show them to log book/insurance docs etc to have bought some time to sort this out before money passed hands. They would have also needed to ensure there was no outstanding finance on the vehicle.
Another question, did you not have a forwarding address for your old address like the service the PO operate???
Good Luck