Show us the original Notice to Keeper (NtK).
What was the date of the alleged contravention and what is the issue date of the NtK? Only those two dates will determine whether the NtK was delivered (given) within the relevant period. The actual date it was received has no relevance whatsoever, unless it can be evidenced (unlikely).
As stated above, you are beyond any appeal stage. It's a pity that you didn't try POPLA if the NtK failed to comply with PoFA.
Ignore Moorside Legal. They are acting as debt collectors. All debt collectors can be safely ignored as they have no power to do anything. They are a third party to any contract allegedly breached by the driver. Ignore, ignore, ignore.
We really don't need to see any debt collection letters.
What you now have to do is wait and see if/when NCP decide to take you to court over the alleged debt. If/when you receive a Letter of Claim (LoC) then come back and show us. You will know it is a real LoC and not a fake one from a debt collector, if it gives you 30 days to pay. All debt collectors will usually only give you 14 days.
A real LoC has to follow the Pre Action Protocols (PAP) and you can check ay letter against paragraph 3 of this document:
Pre-Action Protocols for Debt Claims