Hi Alison,
As I understand it Virgin is a 'cable/fibre' provider for telephone and broadband and BT use copper based technology and do not overlap, where as the majority of other providers piggyback on the BT infrastructure - So I don't think BT will be able to clear anything.
Have you been keeping track of all your attempts at contact and getting this resolved?
Date and Time
No. Phoned (or email sent)
Person spoke to
Discussion / instructions
Time spent
This should help you build you case for poor customer service.
You should be able to look into this online also via the
https://my.virginmedia.com/dashboard/start
Have you found their complaints procedure?
http://www.virginmedia.com/myvirginmedia/contact/complaint-feedback.php
We aim to resolve all written complaints within 28 days and to respond to all e-mails within 48 hours.
As calls from a virgin phone are free could one of your friendly neighbours who have the service let you call from their phone?
Don't rush into signing up for BT as you'll end up with two parallel services (costs).
Good luck
Sparky.