In my experience, if the old line has been taken away or turned off, you have to pay the £120 BT reconnection fee WHOEVER YOU TAKE. We are with Sky but when we moved in here we had to pay BT the £120 to turn on the line (no engineer visit needed but you still pay) and then we immediately transferred across.
For the cheapest deal, have a look at the telecoms pages of moneysavingexpert.com. The advice there is very good.