Advice for Referral/Assessment?

I am currently looking into getting a diagnosis for autism through my GP. I spoke to them over the phone about two weeks ago and they sent over an ASQ for me to fill in - I got 47/50. GP will be referring me, but warned it will probably be a 2-3 year wait for a diagnosis through the NHS. Unfortunately, I need this diagnosis sooner to access support from my university. GP suggested going privately, but I can't afford this. He then shared that some private clinics will charge the NHS instead, so I effectively do not have to pay for assessment etc. However, he could not tell me of any clinics that do this and told me to look this up online/ring around. 

So far, I'm not finding anything useful about this, only really finding information for children. Could be overthinking it, but I am really nervous about phoning up clinics to ask if they offer this service. Does anyone here have any experience of these clinics? Is there any other way to receive a diagnosis quicker? I'm finding my course rather difficult, but cannot access the support I need through the Disability services without proof of an actual diagnosis. Have tried to get funding from my university, but they will not pay for the full amount and I simply can't afford to top up the bill.

Also, any advice from females in academia experiencing autism would be wonderful.

Sorry if my questions have already been answered. I am new to this website and am finding it difficult to find the answers I am looking for. Many thanks.