Driving offences are always personal. so far as the fine/points/ban goes. Nothing to do with your employer. IMO if the delivery driver is booked in at the car park at 12.05 and the offence is committed at 12.00 it would be up to him to prove that it wasn't him driving.
I don't think that the contract start time is relevant. If the delivery driver was delayed say for 2 hours and got a speeding ticket trying to make up the time, then that would be his speeding ticket not yours.
A mate of mine was involved in a speeding offence in similar circs. He couldn't prove that it wasn't him driving, but more to the point, the police couldn't prove that it was so the case got dropped.
I also don't think that signing for the car is relevant or not..whether or not you have signed for the car makes no difference to whether it could have been you driving. I would get the reception to note the time that the keys are dropped off and possibly aslo the time that you collect them, that might help.