I have no idea BOFH, clause 8 in the terms and conditions reads
Upgrades are subject to availability and capacity control at the time of booking. Seats available for this promotion on each flight are limited and may be fully booked during peak periods even though seats are still available in a different booking class in the same cabin. BA accepts no responsibility for an inability to take up this offer if seats in A class are not available on a given date within the Promotional Period.
A shame as they are offering first class tier points, otherwise she wouldn't have been too bothered.
As for cookies, the BA policy is
here. I suppose they could use the session cookie to do what you suggest, but I'd be surprised if they did, but then funny things happen. If you suspect it then just close the browser and the session cookie will clear, or wait 15 minutes and it will clear. I think there is another explanation in that they appear to headline the cheapest fare available, but when it comes to booking the actual flight that fare is not valid. Also the tax/surcharge calculation used to be calculated later and added, but they seem to be putting that into the headline fare now.
The ba website has changed over the years, sometimes it's ahead of the field, sometimes behind. I think that is to be expected. It's got a nice feature in that you can easily compare fares within 14 days of any date you put it. It doesn't always work, I've had problems in the past retrieving bookings on my exec club number. It was doing that a couple of days ago, the missus was unable to do the online checkin, no idea if it is working now.