@ATB your primary concern should be to sort out the V5C issue. Even for a speeding fine, if the address held at DVLA is wrong that's your problem, not that of DVLA or the police. If you don't get a speeding NIP within 14 days because you've failed to get the address updated, you risk 6 points on your licence and the fact that you didn't get the letter at all (let alone within 14 days) would not be a defence.
The council website says the penalty due is £35, this indicates they have issued an informal rejection reoffering the discounted penalty. You therefore need to get the records at DVLA updated urgently, DVLA will record the change of keeper based on the date that you tell them you bought the vehicle, and the council users a system that gives them the RK at the date of the contravention. As long as the DVLA updates their records and backdates the transfer of keepership appropriately, the notice to owner will come directly to you, but you really need to get a wiggle on.
It is also a criminal offence to drive a car on the road if the name and address at DVLA is not correct.