The "pain" of reconnecting your network would involve swapping a couple of Ethernet cables and nothing else.
Reinstalling your modem directly would involve - at most - the addition of the management software or driver.
However, I cannot possibly see how your router could be to blame for the unchanged connection speed. It's mathematically impossible, given the speed of even a slow W-LAN.
I think you should pressure NTL a little more and do let it fob you off.