Hi Ian, thanks for posting here in Microsoft community, it sounds like you've done a thorough job troubleshooting already. Since the issue only occurs at the charity's office and not at your home, it strongly suggests that something in the local environment like the network setup, DNS, or firewall might interfere with Office 365's ability to authenticate or connect to required services.
One possibility is that the ISP or router configuration at the charity's location is blocking or delaying access to Microsoft services.
Office 365 apps often need to validate licenses or sync settings on startup, and if they can't reach the necessary endpoints, they may hang.
Also, have you tried connecting the Dell OptiPlex to a mobile hotspot or different network while at the charity's office to see if Office 365 loads properly in that environment? This could help confirm whether the issue is truly network related.
I hope this helps you out, let me know how it goes.
Chard