I contacted yesterday using the "thinking of leaving Virgin" phone option around 13:00.
Ten minute wait whilst music played. Ok, it was offshore call centre and had to really concentrate as accent wasn;t clear but they dealt with me and renegotiated my package to a much better price than end of contract pricing and the price increase of £4/month.
Even when my partner contacted them on another occasion about their contract, it was a wait of 30 minutes.
Are you using the "leaving option" under change my package on the phone options rather than standard customer services?
Of all the Virgin options, the "leaving Virgin" one usually does work even if you have a wait and I've never had them cut me off. They may have been a bit flippant in response depending who you get and what mood they are in, but never have had them cut me off and I've been with Virgin nearly continuously since 2007.
Contrast to another major electrical retailer named after a certain food, where I've waited 60 mins+ to get a guarantee repair sorted because they didn;t repair properly twice so far....