Note that I don't have any experience with SEN or things like organizing extra time - All my advice can only really be taken as speculation.
From what I'm reading, it was very clear that you needed extra time, and that the school had not offered extra time (Possibly because they didn't know you needed it, possibly because they didn't feel like you needed it). Regardless, I would look to write to your school for GCSE's and A-Levels, explaining the situation to them, asking for a written reference confirming that despite arrangements for extra-time not happening, that it should have been in place, and that you do "qualify" for extra time. If the school itself isn't willing to offer this, write to your previous teacher's asking for the same thing.
This will be your substitute for proof of prior access arrangements; You're in a very unique scenario, and so I'm guessing that the University would have no problem accepting you for extra time if the situation was clearly presented to them. They tend to write blanket statements like "You need proof of prior access arrangements and proof of medical history" to prevent random people applying for extra time unnecessarily, but they tend to make exceptions if it can be shown extra time is necessary (Like here).
As for other questions - Can the university test me, can the university get a psychologist, etc - These are questions best suited to contacting the University directly. Ideally, you should also contact them about not having previous arrangements, or write about this in an "Extra information" space if you have to fill out a form of some sort.
Generally, just contact the University asap. The earlier the better. Most of your questions can be answered that way, and hopefully everything will go well