I also have this issue, and it only happens playing through the Tivo box, so much so I cannot use it.
So I use the following solution:
I got to BBC iPlayer on line
Get to the programme I want
Plugin my Google Chromecast to the TV
And view the program in HD using my phone wirelessly and casting it to the Chromecast!
How can the Tivo box be so poor at streaming, when I can quite happily do it through my phone?? I never had this issue with Sky! This is completely unacceptable, and Virgin Media should be sorting this out, although as I suspect, they have you trapped in a 12 month contract, so do not really care!
It was a terrible service for a full THREE YEARS, but it seems that recently upgrading to their V6 box solved this issue because it has a faster processor. Mind you, Virgin DID NOT offer me the new box that was launched many months ago, I had to contact them and enquire about it.... Hmmm