My WMC7 box runs 2gb ram & 1tb of hdd's. It has an nvidia 9800gt & an old intel cpu - can't remember quite which one, the point being I have a better gpu than yours but half as much memory and in all likelihood a slower cpu, so your system I would guess is better specced for speedy running.
I don't experience your issue, so my educated guess is either input or processing. From 7 years of building windows media centre based systems, I can only think of one occasion when I had to junk a gpu - it was an Nvidia 6300 & no matter what I tried, the thing wouldn't process video smoothly.
So, I reckon if it were my system, I would try either a new input if you're using a digital box or freesat for example - but if you're on Freeview, you can obviously discount that. Otherwise i would possibly suspect the tuner & try to swap that out for another - since it went wrong overnight & presumably the system hasn't applied any updates, I find it unlikely that drivers are the issue. On top of that, Win7 pretty much carries any codec you would need for tv & mainstream video formats, unlike the good old xp days when you had to choose carefully what to download to run video.
So if you decide to rebuild & find yourself back in the same spot, then you have a hardware issue I would contend.