Well, a couple of months later and SP3 downloaded itself automatically, but still came up with 'Access Denied' when I tried to install it.
The Microsoft support site just sent me round in circles, so I gave up and rang them.
2.5 hours on the phone with a very patient and helpful chap and we found the problem - 'registry permissions', whatever those are. Plus I needed to free up some space on the hardrive - although that hadn't been an issue before.
All then installed and re-booted fine. Thanks, Manish!