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If you download and install an email client such as Thunderbird and set it up as an IMAP, not POP, account. This will replicate your folders and download them onto your computer.
Have a read through these links.
Basically it is how you access your email. If you use IMAP it replicates what is on your webmail account which includes all your folders and you can download and save them. If you use POP it will not replicate the folders you have set up on the webmail account and you will not be able to download them.
You could do it manually with POP but it might be very time-consuming if you have lots of folders to keep. Clear the BT Yahoo! Inbox, move all the messages in a folder there, download messages to the email client, move them to a pre-prepared folder; repeat for each folder.