I've got strong doubts as to whether that email is genuine. It sounds like a hoax (possibly intended to get you to reveal your BT account details, so that the account can be hijacked).
The BT Yahoo browser is simply a customised version of Internet Explorer. There's no reason why it should cease to function because of it (allegedly) being no longer supported by BT.
However the vast majority of the 'regulars' around here in AB's Technology section wouldn't touch Internet Explorer (either in its native form or as customised by BT) with the proverbial bargepole. By far the most popular browser here is Firefox:
http://www.mozilla-europe.org/en/firefox/
(It's free. It's easy to download. It's easy to install. It loads pages far faster than IE does. It's generally regarded as more secure than IE. It's also far more flexible because you can customise it to your own preferences with the use of optional add-ons).
Chris