Sounds more like a processing/cpu issue. Network config shouldn't affect windower opening up like that.
Well considering the initial process of Windower(Those dots passing to the right) is it checking for updates, it is more than likely some sort of connection issue. I might be wrong though.
Your IT staff are morons if their solution to a Windows update was to buy new PCs.
As for OP: try a system restore to before the update and try again.
They must have Geek Squad for IT or something. xD
Windows 10 system restore seems much better than predecessors. MS has even been teaching new techs to bury the old wipe and format days and yet I'm guilty of still doing it annually. >.<