If all your emails are presently on the BTMail servers, ie when you check using a web browser they are all there, then you could download and install an email client such as Thunderbird and set it up as POP3. You could then download your emails to that. It should be noted POP3 will only down load the Inbox and not any other folders.
Once downloaded you would have a PST folder on your device which you could then use to back up your emails and transfer to another email account.
The other option is to set up your domain prior to leaving BT and set up an email client such as Thunderbird as IMAP which will show all your folder and emails, and add your Domains email address and your BT email address to it.
Then you can simply drag and drop the emails from your BTMail account into your Domain's email account.
It should be noted that when you leave BT your email account is downgraded to BTMail Basic which is free and can only be used with a web browser.
See link
https://www.bt.com/help/email/bt-email-products
