Introducing wireless repeater or bridge causes double hopping wireless. It greatly reduces speed and efficiency of wireless. I am not 100% sure I understand your arrangement. But there are limited wireless channels and the result of many wireless devices fighting each other is bad.
1. Do you have wireless turned off in the Home Hub? Please do unless it is providing important connections.
2. What mode is the AEBS running in? Because with BT you must use the home hub as a router, you should have the AEBS bridged. Is that the case?
3. Wireless extending networking adds issues as stated. Get rid of this. The best method is to link the TC and AEBS by alternative.. ie ethernet is the best by huge margin. If you cannot do this in your house then use EOP adapters. They are not foolproof but if they work, they keep working with reliability and stabilty and speed that wireless cannot match. You can then run the AEBS and TC as roaming WAP, by using different channels but same SSID (wireless name), and security settings and passwords.