What? If a DPD driver hits you do you not go through DPDs insurance?
The key point is the rider is riding (supposedly) illegally in the OP. I don't know about the T&Cs of the liability insurance, but I'd assume there is no cover if the employee is acting unlawfully and there probably is no legislation of liability.
In the case of a DPD van, the DPD van insurance is liable for third party damages under Road Traffic Act 1988. The van insurance would pay out, but then would seek to recover its costs from DPD as the policyholder for breach of contract (I guess). DPD would then sue the employee for damages.
I'm not sure if the bicycle and any insurance liability would fall under Road Traffic Act 1988, as bicycles don't have to have insurance (unsure about for business use).
I'd like to state I'm not a lawyer and this is my personal understanding