Just to confirm - is your mum the registered keeper of the vehicle, with the V5C document in her name (i.e. not a financed/leased vehicle etc.)? If so, and this is the first notice received, then NCP have sent the notice too late to be able to recover the charge from the keeper, using the provisions of Schedule 4 of the Protection of Freedoms Act. As such, they can only hold the driver liable, and they don't know who that is!
Your mother can therefore appeal online,
as the keeper only, along the lines of the below:
Dear Sirs,
I have received your Parking Charge Notice (Ref: ________) for vehicle registration mark ____ ___, in which you allege that the driver has incurred a parking charge. I note from your correspondence that you are not seeking to hold me liable as the registered keeper, under Schedule 4 of the Protection of Freedoms Act 2012 ("The Act"). You have chosen not to issue a Notice to Keeper in accordance with The Act, and it is now too late for you to do so.
There is no obligation for me to name the driver and I will not be doing so. I am therefore unable to help you further with this matter, and look forward to your confirmation that the charge has been cancelled. If you choose to decline this appeal, you must issue a POPLA code.
Yours,
If appealing online, be careful there are no drop down/tick boxes that cause you to identify who was driving, and keep a close eye on your spam folder for their response. If they do not respond within 28 days, chase them.
If your mum struggles with tech, you can do it for her, but the appeal must be done in her name and address etc.
The arguments around time taken etc. are all perfectly valid, but the above is a much easier approach.