Although you are well aware of why BT ask for a mobile number to verify the account holder @somehairle the part about you not being inconvenienced is clearly not a black and white issue. There's different levels of inconvenience and making any changes to this sort of information, by its very nature, means some inconvenience will be experienced. As already mentioned there are other ways of satisfying the security requirements to make sure its the legitimate account holder wanting to make these changes.
Although there aren't EE/BT stores in every part of the UK, you don't have to run to them and there are still lots of them around. I would have thought a visit to the one that is easiest for you to get to, perhaps done at the same time as big shopping trip to minimise any inconvenience, should be able to get this resolved. If you take plenty of photo IDs eg a passport and/or drivers licence etc along with recent proof of your address eg a few utility bill letters or bank letters etc so the in-store staff can verify you along with a tablet or portable laptop that can be used to access any associated verifying emails, this should be possible, especially if the store management realise the 'usual' methods aren't available to you because you don't have (& nor do you want to have) a mobile/cellular phone.
If that doesn't work then if it was me, I'd send a letter to BT explaining what happened when you tried to get the email address changed in-store and ask them how they'd like you to proceed. There are plenty of people without mobile phones and owning one (or even having access to one) won't be needed to get this resolved.
