Hi wjr and welcome to the forum :flower:
No offense, but your WSUS team is as awake as can be.
You see, there is no free upgrade from Windows 7/8.1 Enterprise to Windows 10 Enterprise and the nag to upgrade only affects the Pro version and only when not handled by WSUS or System Center Configuration Manager.
In other words, if you start to see a nag to update, that nag is an application built by your organization and published to all or only a select group of users in the organization.
Likewise, the actual upgrade is residing on your LAN and is custom built by the IT department. What you may experience as 'suck up bandwidth' is merely a glitch where they forgot to prioritize network bandwidth.
It seems like your organization have finished the licensing part and is about to make the switch to Windows 10. I would not think too much about the fact that you are unable to install it at current as this is a controlled upgrade which will be rolled out in steps based on site, organizational unit, group membership or other policy. There are many reasons for this, one of them is not to overload the support team.
I can not entirely see the point in the nag-app, but it is probably to better serve early birds and give them a feeling of participation.
With more than 28000 clients everything has to be controlled by the IT department, I can imagine that you always get assigned to the nearest printer no matter where you are and so on...
If I were you, I would check the internal website for information as to when the switch is going to happen for your team