I am also from the UK and know the endless waitlists.
Firstly stick with the NHS waiting list you are clearly struggling enough and aware enough to know you need help if the NHS decide its not ADHD they might have alternative help.
If your assessment had been via the NSH and they had said its not ADHD you are entitled to a second opinion.
You never know how experienced the assessor is and some high functioning people need assessing multiple times to catch everything especially if ADHD overlaps with ASD.
Now with your university issues, have you applied for Disabled student allowance (DSA).
As you are awaiting diagnosis and have been told on the NHS route high likely hood of it that is enough to get your university to give you accommodations and if your not already taking to the disabled student department of your university you should do so now. Right now.
In terms of the private diagnosis route you took, if they only asked 5 questions and kicked you out that dosen’t sound like an actual assessment more like a consultation. Did they ask to see old school reports or make you fill in the scale ratings? Did they ask your girlfriend anything?
How thoroughly did you research the person / practice before hand? How long had they been practicing for?
I would right to the practice if if is not just run by the one person who assessed you and point out what you were unhappy with then if you get an diagnosis via the NHS you can have that to back you up.