Well, the car is still showing on the AUC website. Have not been impressed at all with the dealership, and it seems that Harry Fairbairn Irvine are living up to their poor reputation. My experience so far outlined below:
Dealership advertises a car which they've looked after for some 6 years. Although they've looked after the car for a number of years, the dealership sadly loses (!) the service book just at the time they are looking to sell it.
Agree a purchase, subject to being able to confirm the service history which they assure me is complete.
I call BMW GB who send me a written confirmation of all visits for the car to BMW dealerships. The first such visit is in 2010, around 4 years after the car was first registered. This makes me wonder if the car had ever had a running-in service.
The dealer is still happy to sell the car, keeps insisting that the service history was complete and that AUC cars are prepared to the highest standards, etc. Thanks to all the advice on here, I decide to ask the dealership 2-3 times to verify that the warranty would be valid. The first couple of times they insist it is fine, but when I ask the third time, they check with BMW.
Then revert after a few days saying they cannot apply an AUC warranty on the car. I ask for a price to buy the car as is, with the lack of warranty. A couple of days later, I get another email saying they are trying to work something out to get a warranty applied on the car.
A week or so later, they say that they can't put a warranty on it and will sell the car in the trade. Again, I ask for a price at which they sell the car to me (bearing in mind they've had my deposit since December).
Receive an email with a price that represents a discount of a few thousand pounds.
I accept the offer, thinking finally I've sorted out a car, which was important given that I've passed on a few private examples in the meantime.
The dealership then sends a message saying they aren't comfortable selling the car. No other justification beyond a fairly fluffy and vague one-liner.
As far as I'm concerned, I feel they have no integrity whatsoever, and this is one of the worst dealership experiences I've encountered. Half tempted to contact BMW GB to complain, but doubt it will achieve anything.
I now understand why many people don't want anything to do with Harry Fairbairn in Irvine. They should go back to punting cheap cars, which I believe was Arnold Clark's bread and butter.
Rant over!