Wow I wouldn't pay anymore money untill.id driven it. Putting a full MOT on it isn't unreasonable in my opinion, I wonder if there's a reason why they won't? 🤔
If there's no proof of these servicing items then you'll need money off to get these jobs done. Baring in mind you'll be looking £400 minimum for a Cambelt and water pump change.
Most of these warranties aren't often worth the paper they're written on. Unless it's a genuine Audi one by the main dealer.