What happened is going online resulted in everything being redirected to a BT page saying you're not using our DNS and to change settings so you do. Some googling revealed a few people who'd had the same issue and found the (obscure) page that allowed you to undo the block.

I assume they detected it simply by seeing DNS queries going to non-BT servers. Note this was a few years ago when it was pretty common for PC malware to hijack DNS requests, so could be it's changed in the meantime.

N.B. I also recall BT redirecting requests for non-existent domains to some partner of theirs, I assume experimentally as I haven't seen or heard of that for a while.