Kits,
As you taxed it the last time with green slip then surely you would be on their system as the registered keeper ?????
If you are already registered as the vehicle keeper on the DVLA computer and there are no changes to your personal or vehicle details, you may apply for a duplicate by calling 0870 240 0010 using a debit/credit card for payment.
Also as the previous owner has failed to notify them then the charge is waived.
Surely the turn around time on that has got to better than waiting for the V62 to be completed and returned.
Information taken from
here
If not maybe worth asking them for alternative as you are willing to pay etc. and the car is Insured and MOT.