You can fail to do it the easy way or you can manually do it the hard way. The TL;DR is
download and install the MSU file manually.
I started off trying the easy way, letting my Azure VM download it through Windows Update. Despite saying "Downloading", something had clearly finished as TIWorker.exe was causing high CPU usage. I rebooted, for good measure, then used sconfig.cmd to try and download and install the update. I left it running. Overnight. TIWorker.exe used 8 hours of CPU time (about 12 hours elapsed time) before I gave up and killed the process.
I decided to go with Plan B, downloading the MSU file from Microsoft Update Catalog:
http://www.catalog.update.microsoft.com/Search.aspx?q=KB4035631
Download the Server 2016 version and once that file is on your server, simply open it to install the update in a matter of seconds.
This
appears to have been acknowledged in the comments on this Microsoft Partner Support page in the last few days as a known issue allegedly related to the use of Express CAB files that will supposedly be addressed in the coming weeks, and as it's not a security update this isn't a huge deal, but it is annoying and more importantly it may result in increased CPU usage (on a single core) that could affect performance on some servers.
I can definitely understand why many organisations are still using Server 2012 R2 and keeping clear of Server 2016 for now.