Thanks to Through The Light I found out I can use both my broadband connections simultaneously, with a "dual WAN" router.
Well, after a month-long struggle involving two different routers and what feels like dozens of supporters and as many hours of tedious work, it seems I finally got that to work. (It seems for some odd reason NTL did not ask for registration of the new router like it should, so I had to copy the MAC address from the last device that was connected to my cable modem manually into the new Linksys router so it has a "fake" MAC address. This works.)
It seems I have a dual WAN connection now. I only haven't found a clear way to confirm they are both working simultaneously and the load is balanced over both connections. (But I can see they both do work, I can turn them off individually.)
Weirdly, running dual WAN seems to disable my iChat. Most of the time it will not connect now, though sometimes it does. And the account security settings change spontaneously. Very strange. Life with computers: never a dull moment.