Here's what I just did - which might be a superset of what is actually required, because it may be that the 'kill the right
processing folder with fire' is enough, and installing a different QGIS version is not necessary.
1. Upgrade QGIS
After I did that, I noticed that OMBB still was not present in
Processing->QGIS geoalgorithms -> Vector General Tools
First, open the Plugin Manager, and make a note of what version and folder it thinks it's supposed to be using for Processing. The image below is showing the correct version and folder pointer (for my Win7 box, at least).
Originally, the plugin Manager was showing Processing as a different version, in a directory in my folder on
c:\users\me\...\processing - when I examined the folder I noticed its modification date was sometime in 2015, despite the fact that I had literally just finished installing QGIS afresh 5 minutes beforehand. I am quite certain it's not 2015 (I've been wrong before: there was a time a few years ago when it was 2015 for the whole year).
I then got rid of the old directory.
Restarting QGIS afterwards gave a message that processing had trouble loading; a couple of config changes later (changing file pointers for GRASS), everything was smooth again and I was able to do OMBBs for an entire vector layer.
This was the result: