One thing I do like about this setup though, is the fact that the modem is separate from the router. This means I can try out various router configurations without continually resetting my connection (which might cause the line-monitoring systems to lower the speed temporarily to mitigate the perceived drop in line quality).
After a while I became so fed up with my HomeHub 3 that I decided to buy a replacement. I bought a TP-Link TL-WDR3600 on Amazon UK for £45 [*].
Since this router is fairly basic, I have installed the DD-WRT custom firmware on it.
I wrote a guide on doing that which you should check out.
Having upgraded the firmware, I needed to connect my router to my broadband modem. BT's own help page suggests you should use a connection type (or encapsulation) of PPPoA if you want to use your own router with BT Infinity.
Unfortunately the settings that BT suggest do not quite translate to the DD-WRT interface; an encapsulation of PPPoA is not available.
Here are the settings which I got to work (if a setting is not mentioned then it should either be blank or disabled).
Connection Type: PPPoE
Force Reconnect: Enable
Connection Strategy: Keep Alive: Redial Period 30 sec
MTU: Manual 1492
Since this is my main router, I also ensured that its Operating Mode was set to Gateway in Setup -> Advanced Routing.
Once I had saved and applied these settings everything was working again.