BT is a prime example of the maxim that the one thing which communications companies can never do well is to actually communicate with their own customers!
I use a 'business' broadband service, from a company (Claranet) which provides internet facilities to some major multi-national companies (such as BAe Systems). It's not cheap but the service is outstanding.
However, since you've stated that you only get your email from BT, I can see no reason why you can't switch to using GMX:
http://www.gmx.co.uk/
It's free and you can access your mail either via their website or with an email client/app. You can also use the 'mail collector' facility to ensure that any mail sent to your old address is automatically forwarded to your new one.
I use GMX myself and I've been recommending it here for ages, with many ABers deciding to give it a go. All of the feedback I've had has been positive.