Have you got AfterEffects or Combustion? If so the way I do this is to render out as an RPF file and just put all the VT in when you composite, it’s very simple. Basically I’d assign a composting tag to each screen and give them all a different object ID. Remove all the quicktimes and just replace it with a flat colour (black). Remember when you render to RPF to select the options and add ‘object id’
When I then load my render in to Combustion I just select the ‘texture map’ plugin (under 3d) select the object id and assign the relevent quicktime in the ‘layer’ box. Do this for each screen, you will have to render out mattes for each screen to put them all together over the main render.
The best part about this is you can slip and change the clips as you like without having to re-render the 3d constantly