As @Eeeps has already alluded to...
OnDemand programmes are streamed over the internet. Recorded programmes are played from your hard drive.
What you may be doing is creating "bookmarks" for OnDemand programmes, that then appear in your "my shows" list and play directly from the shortcut - but are still OnDemand programmes.
All OnDemand programmes use the same internet connection - can you watch other OnDemand programmes ok? Try watching from, say, ITV CatchUp if you're not sure.
The speed test mentioned, is under Home > Help & Settings > Help > (top of page) > TV Speed test.
I'm a Very Insightful Person, I'm here to share knowledge, I don't work for Virgin Media. Learn more
Have I helped? Click Mark as Helpful Answer or use Kudos to say thanks