Tried some more things:
1. Selected Startup Repair instead of Boot Normally (by accident, my brother was too slow to change it, before then it had defaulted to Boot Normally every time anyway, so it took him by surprise). Anyway, I didn't want to interrupt it if it was doing something, so we let it run but although it said it had identified some problems and was fixing them, after about 10mins it said it had failed and showed
Startup Repair offline
Problem Signature 01 6.1.7600.16385
Problem Signature 02 6.1.7600.16385
05 Auto failover
2. Booted into Safe Mode (without networking) again and uninstalled KB2859537 as this article suggests it can cause BSOD on boot http://news.softpedia.com/news/Microsoft-KB2859537-Update-Causes-BSOD-on-Windows-7-375659.shtml
I'm not sure this was one of the updates that was applied today before it started BSOD, as that article's dated 13 August so I think I'd probably already installed that and it hadn't caused any problems but it seemed worth a shot. It didn't help though.
3. Did a clean boot as per http://support.microsoft.com/kb/929135
but that still resulted in the BSOD.
4. Booted into Safe Mode (without networking) again, went into Device Manager and disabled the onboard Qualcomm Atheros NIC (as it boots in Safe Mode but not in Safe Mode with Networking it seemed worth trying) but that still BSOD as well.
Every time it boots into Safe Mode (without networking), it says "Failed to configure Windows Updates. Reverting" and the first time after uninstalling that update, it said it had to reboot and did so automatically. After that, it still shows that each time but then continues to the desktop. It said that the first time we booted in this mode earlier today as well, before I tried uninstalling that update, but without forcing a reboot.