Can you confirm this issue is with a TiVo and not a V6?
If so, the ‘unplugged cable’ is logical and not physical and suggests that TiVo’s own internet connection (which does not use your broadband) is the source of the problem.
Assuming you have tried a restart of the TiVo (power off then on is the easiest way), you will need to phone in for support or wait here for up to a week for Virgin Media to respond.