Every dojo at which I have trained or visited provides training for newcomers on forward rolls--if not formal beginners' classes, at least an experienced student to coach, and plenty of slow practice rolls before starting in with throws. (A good thing, too. It took me five months to learn to forward roll reliably.)
The fact that a class was exam prep should not affect this. If it was going to be taught above your safe level, you should have been asked not to participate.
Personally, I would leave. The risk of injury is too great, and a severe injury could mean no more aikido ever. Better to look elsewhere, even if it takes a while. Most dojo do a better job than what you're describing.
I hope your head and neck recover promptly.