Luckily, Lenovo has a utility that will let you update all drivers at once, instead of having to hunt for each one. I used the 'System Update' utility and it promptly showed all the drivers that were updated since I got hold of this machine about an year ago; the updates also included a BIOS update. I happily obliged and installed all the updates.
While playing with BIOS before the updates, I had turned off the VT-x, which I remembered, so I went back and after a reboot I switched it back on. But lo and behold, my VirtualBox would not see that setting enabled and would refuse to start any VM that has more than 1 CPU. (It complained VERR_VMX_MSR_LOCKED_OR_DISABLED)
I read on the VirtualBox forum that switching VT-x off and back on worked for one user, so I tried that trick a few times but to no avail. I was about to rollback my BIOS firmware manually (at the risk of bricking my machine) but just then I read this advice in a VMWare post:
Because the VT-x setting is typically locked at power on, it is necessary to fully power down the system after changing any VT-x options in the BIOS. A simple reboot is not sufficient! After saving your BIOS changes, I recommend that you either switch off the power supply itself or pull the power cord(s) out of the wall and wait ten seconds. For laptop systems, you may have to remove the battery as well, although such extreme measures are rarely necessary.
So the trick was to not restart the laptop, but to turn it off, hold your breath for a few seconds and then start it again.
VT-x and my virtual machines have been working fine since then, but my BSOD situation is yet to be evaluated; I hope it works when I go there in a few days.