First (GW - perhaps it's a Mark Hopwood thing) are quite open to it - there was a long series based on their services primarily in the Paddington area a couple of years ago.
GTR, well, they're GTR, aren't they?
I think each TOC will look at how it reflects their 'brand'. GWR makes sense, and any Intercity operator, but imagine GTR if the filming was last year (around Christmas time) when all the problems would be down to them! Great publicity that.
I wouldn't be surprised if GTR actually refused permission to feature them, besides the odd establishing shot or in the background.
It would be perhaps more interesting for a TV documentary to find a way to actually follow a hugely busy commuter operation, like Thameslink, c2c, GA or something. It might actually educate the public that many delays are down to passengers or Network Rail, and could show the domino effect of what seem like mundane, inconsequential, delays that are anything but.
We've had this for the London Underground, but people sort of expect to stand and be squashed on the tube. Once they go above ground, they expect quiet trains with loads of seats.
Imagine if the circumstances were different, if there was a series looking at the Thameslink upgrade work, including the introduction (and ups and downs) of the new Class 700 introduction, explaining the need to change the interior design, a look at the new PIS etc.
It could still (IMO) be done in a way to interest the general public and not just rail geeks.
Perhaps going forward, a production company that might by some luck read this, could consider a future series on Greater Anglia, to look at how the company will replace every single train, introduce new timetables and services etc.