I have never tried an autoland with just one power source available in real life. However the fail operational autopilot is of course able to do just that if the second power source becomes unavailable below Alert Height (200ft AGL). There will be no warning, no problem and a normal autoland and automatic rollout will follow, at least according to the system description and simulator experience. However, by that point the back trim is already in and the second autopilot does not disconnect.
As for starting the approach we have a reference table in our QRH (CAT II/III Requirements), but it is a company list and expressively states that it is for reference only, we have to check the MEL in any case as authoritative document. For CAT II/III operation it says that 2 generators are required (two out of three installed). The same is true for autopilots, two are required.
Interestingly, the MEL only states in the classic section (-300/-400/-500) that both generator busses have to be powered during CAT II/III operation, but not in the NG (-700/-800) section. However it states under the NG section that only one engine driven generator is needed as long as the APU is running throughout the flight which pretty much leads to the same thing (except for ER operation).
I heard stories since my classic days that one autopilot is enough for an autoland, but never tried it as it was a pretty pointless exercise in the middle of europe with alternates available every 10 flight minutes anyway.