I'm sure that there are an increasing number of web sites that don't work too well, or even at all, with IE6. But what's special about BT?
I sort-of gather (I'm not sure he actually said so) that BT is the OP's ISP ... but again, so what? I rarely to never visit my ISP's web site, and I don't give a toss what browsers they might or might not claim that their techies "support" - as far as I'm concerned my ISP provides a pipe down which I send and receive bits, and it's none of their business what the bits are.
If the problem is that the OP finds it important to be able to download his landline phone bill from BT and he won't be able to do so any more soon as they are about to stop supporting his browser then that might (if he cares enough about his phone bill) be a reason for upgrading. But I'm having to make stuff up here, as the information simply wasn't in any of his posts.