I will provide keepers details but it’s difficult to recollect when this happened.
As well as being the Registered Keeper, were you "the person keeping the vehicle" at the time? If so, you have a duty to provide the driver's details. If not you must provide any information that you have which will help identify the driver. So which is it? Either way you must comply with that request. You know when it happened as you've been given the details,
Second named driver was at work in Finchley at that time.
That would seem to make your task of identifying the driver somewhat easier.
Can the speeding fine be challenged with the speed being at the lower limit? What are the successful appeals that can be made for this low limit prosecution?
What lower limit? The limit is 30mph. The car was detected allegedly doing 35mph. 35 is more than 30. if the driver (whoever that was) wants to defend the charge on the basis that he or she was not speeding the court will have to be convinced that the car was travelling at no more than 30mph. However, unless you identify the driver the speeding issue is irrelevant. The police currently do not know who was driving so they cannot take action against anybody for that offence until you tell them.
I see no mileage in the lack of signature either on the NIP or the request for driver's details.
You seem to be intent on converting what will be the offer of a course for the driver into a far more serious outcome for you. If you fail to identify the driver you will face a charge which, if you are convicted, will see a hefty fine, six points and insurance grief for the following five years. So again, what efforts have you made to establish who was driving?