You simply put the TC in bridge mode.. with BT you cannot do authentication on the TC because they use PPPoA.
If as per your profile you are still on Snow Leopard..
First of all plug just the TC lan port into a computer.
Press and hold reset until front led flashes rapidly.
Open airport utility manual setup. Then simply select internet tab and in connection sharing.. off bridge mode.
Update the TC,
You plug ethernet from the TC WAN port to the HH3..
You can set wireless to whatever you like.. or use ethernet.. but it should all work fine.
Having two wireless routers sitting next to each other, it is often best to turn off wireless in one.
Thank you very much for this advice. Seems That I have suceeded in getting this hooked up and currently doing a backup, I guess via the wireless of the HH3 as the TC isn't showing in my list of available networks. However, probably I didn't explain properly, I actually want to use the TC as my router as I used to before I used BT.
Basically full disclosure, my phone socket is in the corner of my house and I can't get wifi through the whole house. I have bought Devolo dLAN that I was intending to place one in the middle of the house and have the TC connected to that. I have successfully managed to set these up that I can connect my macbook to the internet via ethernet cable through these dLAN's so I'm assuming that I could do the same for the TC and use that for internet wifi.
As it stands my TC is showing a green status light but Airport Utility is showing 3 issues
Internet connection - No valid IP address
No DNS servers
I'm a bit worried about trying to fix these as don't want to mess up what you acheived so far, I am at least getting my back ups now so thank you for that. Any pointers to try and resolve these issues please?
You cannot bridge out the home hub unless your ISP uses PPPoE, as that is the only alternative connection the TC allows.. AFAIK BT has never allowed PPPOE.. but check.
Bridge is still the right way to set up the TC, but you can create a wireless network.. do not use extend.. that will mess things up entirely..
Bridge as per screen shot above..
Then create a wireless network.. you can use the same name and security setting as the BT hub.. but set a different channel.. perhaps for initial phases though it works better if you are sure what you are doing and the TC is working.