You could have opened with this - I missed Newsnightthe ComWifi unit is not shown
Unfortunately Nigel, I now have 80% confidence that you're screwed. Remember the forgetting I mentioned earlier? That's the problem. Your comWIFI module is a pretty dumb piece of kit, and for some pretty complex reasons to do with something called ARP, doesn't keep reminding your HomeHub that it's still connected.
To sum up my understanding:-
You've just installed the comWIFI module
Everything works when powered up /rebooted
Some random time later (few hours - few days) comWIFI drops off the network
The same thing happens inside your house as outside using 3G
You have no other oldish ip connected devices in your house like an old cheap ip webcam
You're screwed (again). Plusnet routers had the same problem a year ago, which required a firmware update to resolve. You can't directly fix this. This is now officially a networking issue connected to your HomeHub.
Since I don't like to leave people in a state of screwage, here are some options:-
Pursue the issue through BT's support forums. I would not suggest contacting their support as they're useless. [I hate BT]
Await a HomeHub firmware fix. Good luck.
Swap the router for another. Suggest your Linksys but it may not be allowed onto the BT ADSL.
Swap the comWIFI unit for another. From the little I know about them, they all sound identical, so another model might be smart enough to refresh the network connections. The folks here can probably help with unit selection, I can't.
Long shot - swap the comWIFI for a comWIRED module if such a thing exists. It might be clever enough to refresh the HomeHub.
Very long shot - as above, and interpose a switch between comWIRED and the HomeHub. A switch mind you, not a hub. Switches are designed from the ground up to remember.
Sorry Nigel.