Hi Paul. My experience of buying privately is a lot more recent than yours, but still around 5 years ago. The discounts I negotiated were (from memory) 18% (Hyundai), 21% (VW) and 25% (BMW). All brand new factory order cars. The discounts were achieved as a combination of dealer, manufacturer and finance discounts obtained through taking out a low interest PCP, although I paid them off quite quickly afterwards so didn’t pay any interest and owned the cars outright (you can still keep all the discounts, it’s a loophole).
Given the current chip shortage etc my gut feel would have been that discounts are much harder to come by now, but that doesn’t appear to be the case. A good site to do your research is Drive the Deal, as they quote prices for factory order cars from franchised main dealers (not stock or pre registered cars). I have gone into my local dealerships armed with the percentage discounts quoted there and have always got within a few hundred pounds of those prices.
In addition, discounts of up to 20% off factory options should also be achievable and it is sometimes possible to get servicing plans thrown in or at very reasonable prices. These are also transferable to subsequent owners if you sell the car. Hope that helps.