See below for some reasons why signing up to VM broadband in order to try and rescue a virgin.net mailbox would probably not be a very good idea.
https://community.virginmedia.com/t5/Email/Re-Email-from-Virgin-Media-about-closing-my-email-account/m-p/5577612#M280256
This customer, in similar circumstances, managed to regain temporary access via this approach
https://community.virginmedia.com/t5/Email/virgin-net-email-swindle-Tell-BBC-Rip-Off-Britain/m-p/5586294#M281331
but there are no guarantees you will have similar success as VM's customer service responses are not consistent enough. In any event you should probably speak to ICO as VM deleting mailboxes with no notification is unacceptable IMO. This is particularly so since VM's processes before deletion have been inconsistent with some getting 90 days (as per current T&Cs), some getting 30 days, some getting no notification and some still finding mailboxes have not been deleted at all (even when the customer was actually expecting this).
If you do regain access, see this post from another user in a similar situation with an approach you could try
https://community.virginmedia.com/t5/Email/Another-virgin-net-Cancellation-Victim/m-p/5583582#M281019
another backup method using free software is here
https://community.virginmedia.com/t5/Forum-Archive/Backing-up-your-Emails-One-of-many-solutions/m-p/3539542
Hope you are able to get to some kind of resolution.