The error was seemingly random since over 3000 systems had successfully updated. All were 1607 (though I think this symptom could apply to other versions).
The error was either:
0x80070522: A required privilege is not held by the client
or
0x80240022. Commit Result = 0x00000001
The feature update would not install. Moreover, it failed almost instantly in Software Center. UpdateDeployment.log and WUAHandler.log both reporting 0x80240022.
Cue lots of log-trawling and internet searching (and even an opened case with Microsoft) all without success - shame on you Microsoft! I then noticed that using 'Check online for updates from Microsoft Update' was also immediately failing; usually with error 0x80070006. This led me to this site: https://superuser.com/questions/1195688/windows-update-error-0x80070006
Comparing the sc query wuauserv results for broken and working systems showed a difference in the TYPE:
Broken
I set the process Type to be shared using: sc config wuauserv type= share
Then restart the computer (or you could use sc sidtype wuauserv none if you're doing all this via a remote command prompt using PSEXEC!)
The Feature update then installed successfully from Software Center with no errors.