Firstly, as I was getting myself used to Blender, UV Maps and the Image as Plane function, I modelled a quick TV and using UV Maps to play .mp4s on an image texture plane (I just used an old animation lying about in my folders to test it, so it's not the correct aspect ratio).
This was how the rendered version looked:
The problem was that since I just had the video playing on a plane on top of the TV screen, I couldn't actually move the TV without the plane staying in the one place.
After modelling a second TV, I realised I could simply unwrap the screen's face and use a UV map to play my animation!
The video playing on this TV was a test I had made in Clip Studio Paint before bringing it into After Effects, following a tutorial on how to make a CRT screen effect over existing footage.
The CRT grid ended up being too small and so I couldn't scale it down enough, so this character's features ended up being obscured. In the development phase of the project I'm going to redo this effect and make the grid much larger to preserve details.
Here is the tutorial I followed:
