How do I keep my installed Office when 365 expires?
As you correctly say, your subscription to Office 365 (which is where your "Office 2016" came from) has expired:
Now my sub has expired and any time I open, say, OneNote, I get a popup about needing to renew or enter a key.
Office 365 is only a subscription-based product:
Office 365 is subscription-based. This means that, in the case of most of its subscription packages, it guarantees that you and your team will always have the latest desktop version of all the office software you've subscribed to. Those desktop versions will stop working if you don’t keep paying your subscription fees though.
(emphasis mine)
So that "free-upgrade" can't mean you now own a perpetual license. If you choose to stop paying for the subscription, you can no longer use Office 365, no matter how you got into the product.
Fortunately it sounds like you have a valid license for Office University 2010. If you wish to avoid continuing your Office 365 subscription you should install the product for which you're licensed:
- Uninstall Office 2016 which was provided by your Office 365 subscription.
- Re-install Office University 2010 using the corresponding key.