BBC is probably using HLS, a standard that requires an app by the provider.
The Swedish public TV, SVT, uses HLS as well. They have an app you can download that will enable HLS for svtplay.se.
Send an e-mail to BBC and ask them to do the same.
It is not possible to play all online flash videos with Windows Phone. Flash will not be added to Windows Phone 8 as Adobe is no longer developing the mobile version.
A current (and partial) solution that works on some sites is Metro Browser.