This is my setup, which may be of some assistance. It doesn't do exactly what you ask, but is not too far off.
I have an iMac which acts as an iTunes server. It also has an EyeTV Hybrid and Turbo.264 stick.
The EyeTV is set to record shows and when it has finished recording it immediately exports them to a file that works on Apple TV and adds that export to iTunes. It retains the program data from EyeTV and they generally go into the right place on the TV shows menu.
If I want to keep the program then I use the EyeTV editing function to trim out the adverts and the bits either end. It is really simple to do that, and literally takes a minute or two to edit a show. There is no need to use Handbrake in any of this.
Other advantages of this setup is that we can use the EyeTV app to watch recordings or live TV on our iPads, including remotely over the internet.
It is a great setup, but there are a couple of problems:
You have to wait for a program to finish and then export to Apple TV format before you can watch them. On average it takes about half the program length to export (on a 2010 iMac), so this can be a pain.
It can also be a bit of a pain to manage the programs. You end up with two copies of each program (one on the EyeTV library and another in iTunes) and you have to go to the Mac to delete them.
We also have Sky+ and we now tend to use that for general TV watching and recording regular programs. I record on the EyeTV if I think there is a chance of us wanting to keep the program and as a backup (which I don't export to iTunes) in case the Sky box has problems. If you want to use EyeTV for your general TV watching then you really need to be quite organised. The main benefit that Sky has over EyeTV, on top of being able to watch live, is that you can delete the program straight away after watching without having to go to the computer. We found that watched shows were building up and as I ended up clearing them off there was a danger of me deleting something my wife wanted to keep.
One thing I did was, when ripping my DVD's I added the letter Z to the beginning of the show sort field in iTunes. This means that any shows that are recorded on the EyeTV show up at the top of the TV show list on the Apple TV so it's easy to see what has recorded.
As for your specific requirements, I'm not sure whether there are settings that will automatically put the iTunes copy on your NAS, but I don't see why not. I'm sure that there is much more scope for automation as the original gets into iTunes automatically anyway. You obviously need human input to get rid of adverts but that is very simple.
I think you should be able to find a system that works, as long as you keep your recorded shows organised.