Interestingly, I was also using an RJ45 wall socket wired to the router. It was too far away to try direct but oddly Wi-fi works using a wireless access point connected to the socket.
Also try reserving the IP address at the router and enabling iPv6 if supported.
Has anyone tried the new BT Smart Hub 2, does this resolve the issue?
I've had the issue on both Xbox One S and Xbox One X. The router log in simple terms is showing as if the network cable for the Xbox is quickly being pulled out and re-inserted every couple of seconds.
Oddly, the connection mostly remains, however during game downloads I notice the speeds can drop down to KB levels and are very erratic in between.
If you have a direct connection from the xbox to the home hub, then there may be some form of incompatibility between the interfaces. This can vary between different home hubs, owing to manufacturing tolerances.
This can usually be overcome by adding an Ethernet switch between the LAN port of the home hub, and the xbox.